Science, Faculty of
Mathematics, Department of
DSpace
UBCV
Hong, Han
Doctor of Philosophy - PhD
University of British Columbia
In this work, we focus on three problems. First, we give a relationship between the number of eigenvalues of the Jacobi operator below a certain threshold and the topology of closed constant mean curvature (CMC) surfaces in three-dimensional Riemannian manifolds. We then obtain that the (weak) Morse index of CMC surfaces in an arbitrary 3-manifold is bounded below by a linear function of the genus when the constant mean curvature is greater than a certain nonnegative value. In particular, this implies that stable CMC surfaces are topological spheres. Corresponding results for CMC surfaces with free boundary in 3-manifolds with boundary are obtained as well. Second, we consider the space of embedded free boundary CMC surfaces with bounded topology, bounded area, and bounded boundary length in a 3-manifold N with boundary. We show that this space is almost compact in the sense that any sequence of surfaces in this space has a convergent subsequence that converges to a free boundary CMC surface, graphically and smoothly except on a finite set of singularities. If in addition Ric_N>0 and the boundary of N is convex, then the convergence is at most 2-sheeted. In particular, it is 1-sheeted if the limiting surface is not a minimal surface. Third, we consider the maximization of Steklov eigenvalues in higher dimensions. We show that for compact manifolds of dimension at least 3 with nonempty boundary, we can modify the manifold by performing surgeries of codimension 2 or higher, while keeping the Steklov spectrum nearly unchanged. This shows that certain changes in the topology of a domain do not have an effect when considering shape optimization questions for Steklov eigenvalues in dimension 3 and higher.
Index estimates and compactness for constant meancurvature surfaces and Steklov eigenvaluesbyHan HongA THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDCTORAL STUDIES(Mathematics)The University of British Columbia(Vancouver)May 2021© Han Hong, 2021The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Index estimates and compactness for constant mean curvature surfacesand Steklov eigenvaluessubmitted by Han Hong in partial fulfillment of the requirements for the degree ofDoctor of Philosophy in Mathematics.Examining Committee:Jingyi Chen, Professor, Mathematics, UBCSupervisorAilana Fraser, Professor, Mathematics, UBCCo-supervisorFei Zhou, Professor, Physics & Astronomy, UBCUniversity ExaminerRichard Froese, Professor, Mathematics, UBCUniversity ExaminerMartin Man-chun Li, Associate Professor, Mathematics, CUHKExternal ExaminerAdditional Supervisory Committee Members:Albert Chau, Professor, Mathematics, UBCSupervisory Committee MemberiiAbstractIn this work, we focus on three problems. First, we give a relationship betweenthe number of eigenvalues of the Jacobi operator below a certain threshold and thetopology of closed constant mean curvature (CMC) surfaces in three-dimensionalRiemannian manifolds. We then obtain that the (weak) Morse index of CMC sur-faces in an arbitrary 3-manifold is bounded below by a linear function of the genuswhen the constant mean curvature is greater than a certain nonnegative value. Inparticular, this implies that stable CMC surfaces are topological spheres. Corre-sponding results for CMC surfaces with free boundary in 3-manifolds with bound-ary are obtained as well. Second, we consider the space of embedded free bound-ary CMC surfaces with bounded topology, bounded area, and bounded boundarylength in a 3-manifold N with boundary. We show that this space is almost com-pact in the sense that any sequence of surfaces in this space has a convergent subse-quence that converges to a free boundary CMC surface, graphically and smoothlyexcept on a finite set of singularities. If in addition RicN > 0 and the boundaryof N is convex, then the convergence is at most 2-sheeted. In particular, it is 1-sheeted if the limiting surface is not a minimal surface. Third, we consider themaximization of Steklov eigenvalues in higher dimensions. We show that for com-pact manifolds of dimension at least 3 with nonempty boundary, we can modifythe manifold by performing surgeries of codimension 2 or higher, while keepingthe Steklov spectrum nearly unchanged. This shows that certain changes in thetopology of a domain do not have an effect when considering shape optimizationquestions for Steklov eigenvalues in dimension 3 and higher.iiiLay SummaryCMC surfaces are critical of area functional for compactly supported variations thatpreserve the enclosed volume. Free boundary CMC surfaces are a special class ofCMC surfaces that meet the boundary of the manifold orthogonally.The Morse index is defined to be the number of "directions" in which one canperturb the surface to decrease its area up to second order. First, we relate thetopology of a CMC surface with its Morse index. Second, we show that the spaceof free boundary CMC surfaces of 3-manifolds with bounded topology and meancurvature is weakly compact.The Steklov eigenvalue is an important topic in spectral geometry and geo-metric analysis. Third, we show that certain aspects of the topology of a domaindo not have an effect when considering maximization of Steklov eigenvalues indimensions 3 or higher, in contrast to the case in dimension 2.ivPrefaceThis thesis is based on three works, one of them has been published in an academicjournal, and the other two are submitted for publication and under review.The content discussed in Chapter 2 is based on the paper "Index estimates forsurfaces with constant mean curvature in 3-dimensional manifolds" [4] publishedin the journal Calculus of Variations and Partial Differential Equations, Volume60, 3 (2021), pages 1-20. This is a joint work with Nicolau S. Aiex under theguidance of my supervisors, Jingyi Chen and Ailana Fraser.The content in Chapter 3 is based on the paper "Compactness of the space offree boundary CMC surfaces with bounded topology" [3], which has been submit-ted to an academic journal. This was a joint work with Nicolau S. Aiex under theguidance of my supervisors, Jingyi Chen and Ailana Fraser.The content in Chapter 4 is based on the paper "Higher dimensional surgeryand Steklov eigenvalues" [39], which has been submitted to an academic journal.I chose this problem under guidance of my supervisors, Jingyi Chen and AilanaFraser, and was responsible for all aspects of this work.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Index estimates for CMC surfaces in 3-dimensional Riemannianmanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Compactness of the space of free boundary CMC surfaces withbounded topology . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Higher dimensional surgery and higher Steklov eigenvalues . . . . 71.4 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Index Estimates for CMC Surfaces in 3-Dimensional RiemannianManifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Preliminaries on CMC Surfaces and Morse Index . . . . . . . . . 152.3 Calculations and Main Results . . . . . . . . . . . . . . . . . . . 18vi2.4 Applications in the Classification of Stable CMC Surfaces and Isoperi-metric Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Compactness of the Space of Free Boundary CMC Surfaces withBounded Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2 Preliminaries on Convergence of Surfaces . . . . . . . . . . . . . 413.3 Curvature Estimate for Free Boundary CMC Surfaces . . . . . . . 453.4 Removable Singularities . . . . . . . . . . . . . . . . . . . . . . 473.5 Compactness Theorem . . . . . . . . . . . . . . . . . . . . . . . 524 Higher Dimensional Surgery and Steklov Eigenvalues . . . . . . . . 584.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2 Continuity of Steklov Eigenvalues under Codimension 2 Surgeries 604.3 Some Results on Shape Optimization for Steklov Eigenvalues in Rn 72Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82viiList of FiguresFigure 3.1 Blow up picture of a free boundary CMC surface near singu-larity on the boundary . . . . . . . . . . . . . . . . . . . . . 48Figure 4.1 The construction of the contractible domain Bnε,δ in Rn for nâ¥3: δ is the radius of the neck and ε is the radius of the innerball, and we delete the neck and inner ball from the unit ball. . 73Figure 4.2 The construction of â¦nÎµ,l in Rn, nâ¥ 2: l overlapping unit balls. 74Figure 4.3 Limit of â¦nÎµ,l in Rn, nâ¥ 2, as ε â 0. . . . . . . . . . . . . . . 74viiiAcknowledgmentsAbove all others, I would like to thank my supervisors Jingyi Chen and AilanaFraser for their support and encouragement during the entire PhD program. Atmany stages in the course of my research pursuit in geometry I have benefited alot from their advice and discussions. Their positive outlook and perceptions inresearch inspired me. I am also grateful for their editing to the production of thisthesis.I would also like to thank Kalle Karu, who has acted as my teaching instructorin charge when I taught at UBC. He played an important role in my path to aprofessional teacher. I am very thankful for his generosity in helping me to startmy teaching career. I would also like to thank all staff in the math department whohave offered me help.I would like to thank the member of supervisor committee, Albert Chau, for hishelp in the PhD candidate oral defence and his advice. I would also like to thankRichard Froese for his generous help in my PhD candidate oral defence.I would like to thank my research partner, Nicolau Aiex for his help and adviceto me. I would also like to thank my friends and colleagues in math department forall of the support and useful discussions. Finally, I would like to thank my parentswho have been so supportive during my overseas graduate career. I would not havemade it so far without their continuous encouragement.ixChapter 1IntroductionThis work is dedicated to problems related to constant mean curvature (CMC) sur-faces and Steklov eigenvalues.CMC surfaces are critical points of the area functional with respect to varia-tions that preserve the enclosed volume, and satisfy certain nonlinear partial differ-ential equations. Classical examples of CMC surfaces in R3 include the Delaunaysurfaces, which are surfaces of revolution with constant mean curvature. Explic-itly, these are the cylinder, the sphere, the catenoid, the unduloid and the nodoid.CMC surfaces are important objects in isoperimetric problems. In particular, theclassical isoperimetric problem in R3 states that spheres have least area among allsurfaces enclosing a fixed volume. In a compact Riemannian manifold M thereexists a compact region whose boundary minimizes area among all regions withfixed volume. Moreover, when the dimension is less than or equal to seven, theboundary of any minimizing region is a smooth embedded hypersurface with con-stant mean curvature, and if M has nonempty boundary, then the boundary of theCMC hypersurface meets the boundary of M orthogonally.In addition to problems in the theory of CMC surfaces, we study problemsconcerning the maximization of Steklov eigenvalues. The Steklov problem is animportant and much studied eigenvalue problem. The study of Steklov eigenval-ues, which can be seen as the spectrum of the Dirichlet-to-Neumann operator, hasimportant applications in electrical impedance tomography that is used in medicaland geophysical imaging. Just as for the classical Dirichlet and Neumann Lapla-1cian eigenvalues, the shape optimization of Steklov eigenvalues has attracted a lotof attention, especially in recent years. The relationship between the existence ofmaximizing metrics of Steklov eigenvalues on surfaces and the existence of freeboundary minimal surfaces of the unit ball in R3 was estabilished by Fraser andSchoen [30]. The existence of maximizing metrics for Steklov eigenvalues onsurfaces, which can be considered as a maximization problem among all smoothmetrics, has been used to construct free boundary minimal surfaces in the unit ball[24, 27, 28, 30]. In chapter 4 we study shape optimization for Steklov eigenvaluesin higher dimension.More specifically, we focus on three problems: finding index bounds for closedCMC surfaces and free boundary CMC surfaces in general three-dimensional Rie-mannian manifolds, proving compactness of the space of free boundary CMC sur-faces with bounded topology in 3-manifolds, and showing that the certain changesin the topology of a Riemannian manifold of dimension 3 or higher do not havean effect when considering shape optimization for Steklov eigenvalues. Here, weoutline the problems and state the main results, and outline the layout of the thesis.1.1 Index estimates for CMC surfaces in 3-dimensionalRiemannian manifoldsIf M is a three-dimensional Riemannian manifold with nonempty boundary and Σis a two-dimensional Riemannian surface with nonempty boundary âÎ£â âM, thenΣ is a free boundary surface with constant mean curvature (CMC) if it is a criticalpoint for area functional with respect to all variations that leave the boundary ofΣ in the boundary of M and preserve the enclosed volume of Σ (see definition forenclosed volume of CMC surfaces with boundary in Section 2.2). By the firstvariation formula, Σ is a free boundary CMC surface if and only if it has constantmean curvature and its boundary meets the boundary of M orthogonally. When theconstant mean curvature is zero, Σ is a minimal surface. If Σ is a closed surface,we have similar concepts by ignoring the boundary. Note that CMC surfaces withnonzero mean curvature are always two-sided since the unit normal vector fieldinduces a trivialization of the normal bundle.Considering the second derivative of area functional, we obtain a quadratic2form Q that acts on smooth functions on the surface. We are interested in theweak Morse index of this form, when it is restricted to the function space S = {u âCâ(Î£) :â«Î£ u = 0}. Roughly speaking, the weak Morse index of a CMC surfacemeasures the number of independent directions in which one can perturb the sur-face to decrease its area to the second order without changing the enclosed volume.More specifically, the weak index is the maximal dimension of a subspace of S suchthat Q(u,u)< 0 for all uâ S{0}. Here Q(u,u) = â«Î£ |âu|2â(|A|2+RicM(Î½ ,ν))u2where |A| is the length of the second fundamental form and RicM(ν ,ν) is the Riccicurvature of the ambient manifold M evaluated on the surface in the direction ofthe unit normal ν .In [16], Cavalcante and de Oliveira obtained that the weak index of a closedCMC surface in a simply-connected space form with constant curvature c = 0 orc = 1 is bounded from below in terms of the genus g of the surface, i.e.,Indw(Î£)â¥ g3+ c .Their work was inspired by a paper of Ros [47]. To obtain their result, theyfirst considered a new associated operator LË : Sâ S (see [10]) given byLËu = Luââ«Î£Lu|Î£|where L is the usual Jacobi operator L = â+(|A|2 + 2c). The weak index is thenumber of negative eigenvalues of the operator LË. Their result follows by compar-ing the eigenvalues of the Hodge Laplacian â[1] and the eigenvalues of the operatorLË. Corresponding results for free boundary CMC surfaces in mean convex subdo-mains of R3 and S3 were obtained in a similar manner in [15].We generalize the results in [15, 16] to CMC surfaces in general 3-manifolds.Unlike the case of space forms, our surfaces sit in 3-manifolds in which we donâthave global coordinates of harmonic one-forms to use as test functions in the sec-ond variation formula. However, with aid of the Nash embedding theorem, we canembed the ambient 3-manifold into Euclidean space Rd , of some dimension d, andobtain the following theorem.Theorem 1.1.1. Let (M,g) be a 3-dimensional manifold without boundary isomet-3rically embedded in Rd and Σ a closed orientable immersed CMC surface in Mwith HΣ 6= 0. Assume there exist a real number Ï and a q-dimensional vector spaceVq of harmonic vector fields on Σ such that any non-zero ξ â Vq satisfiesâ«Î£2âi=1(|IIM(ei,Î¾ )|2+ |IIM(ei,?ξ )|2)â (RM +H2Σ)|ξ |2 dVΣ < 2Ï â«Î£|Î¾ |2 dVΣ.Then#{eigenvalues of LËÎ£ that are strictly smaller than Ï} â¥ q2d .By setting Ï = 0, we are able to get weak index estimates.Corollary 1.1.2. With the same setting as in Theorem 2.3.5 . Suppose that everynon-zero ξ âH 1(Σ) satisfiesâ«Î£2âi=1(|IIM(ei,Î¾ )|2+ |IIM(ei,?ξ )|2)âRM|Î¾ |2 dVΣ < â«Î£H2Î£|Î¾ |2 dVΣ. (1.1.1)ThenIndex(Î£)â¥ gd.Note that in dimension two, the dimension of space of harmonic vector fieldsis equal to the first Betti number, i.e., q = 2g.The left-hand side of assumption (1.1.1) only involves the second fundamentalform Î M of the ambient manifold M embedded in Rd and the scalar curvature RMof M. There are a number of three manifolds that satisfy condition (1.1.1), such asR3 and S3. One application of Corollary 1.1.2 is:Corollary 1.1.3. Let M be a closed Riemannian manifold of dimension 3. Thereexist constants C > 0 and H0 â¥ 0 depending on M such that every closed immersedCMC surface Σ of genus g in M and mean curvature |HΣ|> H0 satisfiesIndex(Î£)â¥Cg.In particular, if Σ is stable then it must be a sphere.Corollary 1.1.3 proves a CMC version of a conjecture of Schoen and Marques-Neves [44] for surfaces of sufficiently large constant mean curvature in arbitrary43-manifolds. Comparing it to the conjecture of Schoen and Marques-Neves, thisresult has no condition on the curvature of the ambient 3-manifold. We also provedsimilar results for CMC surfaces with free boundary in general 3-manifolds.1.2 Compactness of the space of free boundary CMCsurfaces with bounded topologyWe are also interested in the space of free boundary CMC surfaces possessingcertain properties in general compact three-dimensional manifolds with boundary,such as the space of free boundary CMC surfaces with fixed genus and the numberof boundary components in the unit ball in R3. It is an interesting open questionwhether this space has only finitely many elements up to congruence. We showthat this space is compact in a weak sense.The study of compactness of the space of minimal surfaces stems from thework by Choi and Schoen [19]. More specifically, they showed that the space ofclosed embedded minimal surfaces of fixed topological type in a three-dimensionalmanifold with positive Ricci curvature is compact in the Ck topology for any kâ¥ 2.As a consequence, a global curvature estimate for closed embedded minimal sur-faces in terms of their genus was obtained. This compactness result was then gen-eralized by Fraser and Li [26] to the space of free boundary embedded minimalsurfaces of fixed topological type in compact three-dimensional manifold with pos-itive Ricci curvature and strictly convex boundary. Recently, Sun [55] consideredsurfaces with constant mean curvature and proved a compactness result for thespace of closed embedded CMC surfaces of fixed topological type in a compactthree-dimensional manifold with positive Ricci curvature. Here, the definition ofcompactness is weaker than that defined in Choi-Schoenâs work due to the appear-ance of a neck pinching phenomenon. In Chapter 3 we prove a compactness resultin the case when the surfaces are free boundary CMC surfaces.To prove compactness it is equivalent to show that any sequence of surfaces hasa subsequence that converges to a limiting smooth surface locally, graphically, andsmoothly. The idea for proving such convergence for surfaces with bounded topol-ogy originates from [19]. The key ingredients and steps are as follows: (1) Finitearea: positive ambient Ricci curvature gives lower bound on the first Laplace eigen-5value of a minimal surface, this together with a topological upper bound on the firstnormalized Laplace eigenvalue shows that bounded topology implies boundednessof the area of the sequence of closed embedded minimal surfaces. (2) Curvatureestimate: bounded topology, combined with the Gauss-Bonnent theorem and thearea bound obtained from (1) implies finite total curvature; that is, the L2 norm ofthe second fundamental form of the surface is finite. Using a scaling argument,the L2 estimate implies a local Lâ estimate. (3) Removable singularity: boundedtopology of the minimal surfaces and classical harmonic function theory impliesthat isolated singularities are removable. (4) Multiplicity one: eigenvalue estimatescan be used to show that the convergence is of multiplicity one. Finally the Arzela-Ascoli theorem and Allardâs regularity can be used to show the desired smoothconvergence. When the minimal surfaces have free boundary in an ambient mani-fold with convex boundary, Fraser and Li [26] used estimates for the first Stekloveigenvalue and an isoperimetric inequality for minimal surfaces to obtain an areabound as in step (1). The result follows by making appropriate adjustments to theother steps.In the case of free boundary embedded CMC surfaces, the situation becomesmore complicated. First, we donât have a lower bound on Steklov eigenvalues forCMC surfaces with boundary. We also lack corresponding isoperimetric inequali-ties. Thus we additionally assume that the sequence of surfaces has bounded areaand perimeter. We then apply a standard blow-up argument to show that finite totalcurvature implies pointwise bounded second fundamental form. Second, since thesurface is not minimal, we canât use harmonic function theory to prove that singu-larities are removable. However, we can use a blow-up argument to show that thetangent cone at a boundary singular point is a half plane, and as a consequence, thispoint can be filled in to the closure so that the surface is smooth up to the bound-ary. The interior removable singularity theorem can be proved in a similar manner.Last, we donât have Steklov eigenvalue estimates to prove that the multiplicity ofconvergence is one. Nevertheless, we are still able to prove this, since if the mul-tiplicity of convergence is not one, we can construct a positive Jacobi field on thelimiting surface, which implies strong stability. However, when the ambient spacehas positive Ricci curvature and convex boundary, no free boundary CMC surfacecan be stable, which thus leads to a contradiction.6More specifically, we prove the following compactness result.Theorem 1.2.1. Let N be a compact 3-dimensional manifold with boundary. Sup-pose HâN â¥ H0 > 0 and let Σi be a sequence of free boundary embedded CMCsurfaces with mean curvature Hi, genus gi and number of ends ri satisfying:(a) |Hi| â¤ H0;(b) gi â¤ g0;(c) ri â¤ r0;(d) area(Î£i)â¤ A0 and(e) length(âÎ£i)â¤ L0.Then there exists a smooth properly almost embedded CMC surface Σ â Nand a finite set Î â Σ such that, up to a subsequence, Σi converges to Σ locallygraphically in the Ck topology on compact sets of N \u00CE for all k â¥ 2. Moreover, ifΣ is minimal then it is properly embedded.If in addition (N,âN) satisfies either RicN > 0 and AâN â¥ 0 or RicN â¥ 0 andAâN > 0, then:(i) when HΣ = 0 the convergence is at most 2-sheeted;(ii) when HΣ 6= 0, then the convergence is 1-sheeted away from Î.The assumption that the mean curvature of the boundary of the ambient man-ifold is greater than that of the sequence of surfaces guarantees proper embe-deddness.1.3 Higher dimensional surgery and higher StekloveigenvaluesAnother topic closely related to minimal surfaces is the Steklov eigenvalue problemon two-dimensional surfaces. Fraser and Schoen [29, 30] developed an existencetheory for maximizing metrics on surfaces of genus zero and establised a connec-tion between the existence of maximizing metrics for normalized Steklov eigen-values and the existence of free boundary minimal surfaces of the unit ball in Rn.7They showed that a metric that maximizes the k-th normalized Steklov eigenvaluearises geometrically as the induced metric on a free boundary minimal surface of aEuclidean ball, by showing that one can construct a conformal minimal immersioninto a ball from the eigenfunctions corresponding to the Steklov eigenvalue.Our third problem will focus on the maximization for Steklov eigenvalues inhigher dimensions. The theory of maximization for Steklov eigenvalues startedfrom the theorem of Weinstock [57] which states that the unit disc uniquely maxi-mizes the first nonzero Steklov eigenvalue among simply-connected domains withfixed boundary length 2pi in R2. An interesting open question in higher dimen-sions is on which domain (or in the limit of which sequence of domains) withfixed boundary volume in Rn for n â¥ 3 the first Steklov eigenvalue is maximized.Bucur, Ferone, Nitsch and Trombetti proved in [13] that the unit ball imaximizesthe first normalized Steklov eigenvalue in the class of convex domains in Rn forn â¥ 3. On the other hand, Fraser and Schoen [31] showed that the unit ball is nolonger the maximizer in the wider class of contractible domains in Rn for n â¥ 3.We generalized Fraser and Schoenâs result to higher eigenvalues:Theorem 1.3.1. Let nâ¥ 3. For any k â¥ 1, there exist a contractible domain â¦â inRn such thatÏÂ¯ j(â¦â)> ÏÂ¯ j(Bn)for j = 1, · · · ,k.This result is a consequence of an interesting result on continuity of the spec-trum under certain higher dimensional surgeries. It was shown in [31] that the num-ber of boundary components of a compact manifold does not affect the supremumof the normalized Steklov eigenvalues. We prove more generally that the supre-mum of the normalized Steklov eigenvalues among all manifolds is the same asthe supremum among manifolds with relatively simple topology. This is achievedby studying higher dimensional surgeries up to codimension 2, in the followingtheorem. If Σ is a compact properly embedded m-dimensional submanifold of â¦,given δ > 0 small, we denote by â¦Î£,Î´ the Lipschitz domain obtained by removingthe δ tubular neighbourhood of Σ from â¦.Theorem 1.3.2. The Steklov spectrum of a compact Riemannian manifold with8boundary changes continuously under surgeries of codimension at least two, in thesense that, if nâmâ¥ 2 thenlimÎ´â0+ÏÂ¯ j(â¦Î£,Î´ ) = ÏÂ¯ j(â¦)for j = 0,1,2, . . ..Here ÏÂ¯ j(â¦) = Ï j(â¦)|ââ¦|1/nâ1 denotes the normalized j-th Steklov eigen-value. Using a similar idea, we also show that in Rn, for nâ¥ 3, the j-th normalizedSteklov eigenvalue is not maximized in the limit by a sequence of contractible do-mains degenerating to the disjoint union of j unit balls, in contrast to the case indimension two [34].1.4 LayoutThe focus of Chapter 2 is the proof of Theorem 1.1.3. We first introduce the prob-lem by providing an overview of closed CMC surfaces and free boundary CMCsurfaces as well as the definition of the Hodge Laplacian. We also provide all ofthe necessary calculations that are used to prove main theorem as well as its appli-cations.In Chapter 3, we concentrate on proving Theorem 1.2.1. We first explain that apointwise curvature bound implies local smooth convergence and prove a curvatureestimate and a removable singularity theorem. Finally, we conclude the chapter byproviding a detailed proof of Theorem 1.2.1.Chapter 4 is dedicated to proving Theorem 1.3.2 and Theorem 1.3.1. Weintroduce the Steklov eigenvalue problem and prove an L2-noncentration lemmafor Steklov eigenfunctions. We then prove continuity of Steklov spectrum underhigher dimensional surgeries and apply this to prove Theorem 1.3.1.9Chapter 2Index Estimates for CMCSurfaces in 3-DimensionalRiemannian Manifolds2.1 IntroductionA closed hypersurface of constant mean curvature (CMC) may be variationallycharacterized as a critical point of the area functional under variations that preservethe enclosed volume. In a similar way, a free boundary constant mean curvature(free boundary CMC) hypersurface is a critical point where, in addition, the bound-ary is restricted to lie in a closed hypersurface. If such a hypersurface minimizesarea for small perturbations then it is said to be stable for the corresponding prob-lem. For example, solutions to the isoperimetric problem, that is, the problem offinding a hypersurface, with or without boundary, that has least area for a fixedenclosed volume, are stable CMC hypersurfaces.In [11, 12] Barbosa-do Carmo and Barbosa-do Carmo-Eschenburg classifiedstable closed CMC hypersurfaces of Euclidean spaces, spheres and hyperbolicspaces. A similar result was obtained by Souam [53] for stable free boundaryCMC hypersurfaces in a hemisphere and more recently for closed CMC surfacesin S2ÃR and H2ÃR. Other classification results for stable free boundary CMC10surfaces were obtained by Ros-Vergasta [48] and later improved by Nunes [45].It is also natural to study CMC hypersurfaces of higher index. That is, those thathave some small pertubations that decrease area with fixed enclosed volume.In [56] Torralbo-Urbano made use of isometric embeddings of homogenous3-manifolds into Euclidean space to study stable closed CMC surfaces and theisoperimetric problem in Berger spheres. The proof uses coordinates of hamornicvector fields to construct test functions for the second variation of area.In the case of minimal surfaces, there have been multiple results establishinga connection between the topology of the surface and its index. For example, doCarmo-Peng [23], Fischer-Colbrie-Schoen [25] and Pogorelov [46] independentlyproved that stable two-sided minimal surfaces in R3 are planes.In [47] Ros proved that the index of a minimal surface in R3 is bounded be-low by a linear function of its genus. This was later extended and improved byChodosh-Maximo [17, 18]. A corresponding result was shown by Savo [50] forminimal hypersurfaces in n-spheres. In both situations the authors use harmonicvector fields to construct test functions but their constructions are fundamentallydifferent. Ros uses the coordinates of harmonic vector fields on the surface withrespect to the usual basis of R3, which works seamlessly since the ambient space isflat. On the other hand, Savo uses the coordinates of a bivector involving harmonicone-forms and later takes the average of the second variation for all such test func-tions. This allows him to overcome the extra term in the second variation given bythe curvature of the ambient space.Using a similar method to Savo [50], Sargent [49] proved corresponding indexestimates for free boundary minimal hypersurfaces in convex bodies of Euclideanspace. Ambrozio-Carlotto-Sharp [7] used both Rosâ and Savoâs approaches to re-late the index of a closed minimal hypersurface in an arbitrary ambient manifoldthat can be suitably embedded into some Euclidean space to its first betti num-ber. The authors later did the same for free boundary minimal hypersurfaces in2-convex domains [8].Our main results in this chapter follow the natural generalization of Torralbo-Urbano for higher index CMC surfaces, similar to Ambrozio-Carlotto-Sharpâs ap-proach. The main difference is that we are only allowed to use admissible func-tions. Fortunately, the coordinates of harmonic vector fields are admissible in the11closed case. In the free boundary case we need the extra condition that these vec-tor fields are tangential along the boundary. In either case we have to use boththe harmonic vector field and its Hodge dual, so it only makes sense in the caseof orientable CMC surfaces. It is not clear whether or not Savoâs test functionspreserve enclosed volume, so a generalization to higher dimensions doesnât seemto be straightforward.A surprising difference between the CMC case and the minimal case, at leastin the case of surfaces, is that the extra term involving the non-zero mean curva-ture allows us to have a wider variety of applications. More specifically, some ofthe ambient spaces that satisfy the conditions of our theorems for non-zero meancurvature do not work in the case of minimal surfaces, for example, flat ambientspaces. The same results in the case of zero mean curvature can be found in [7, 8],so we restrict our statements to the particular case when the mean curvature isconstant and non-zero.Our main theorem for closed orientable CMC surfaces is:Theorem 2.3.5. Let (M,g) be a 3-dimensional manifold without boundary isomet-rically embedded in Rd and Σ a closed orientable immersed CMC surface in Mwith HΣ 6= 0. Assume there exist a real number Ï and a q-dimensional vector spaceVq of harmonic vector fields on Σ such that any non-zero ξ â Vq satisfiesâ«Î£2âi=1(|IIM(ei,Î¾ )|2+ |IIM(ei,?ξ )|2)â (RM +H2Σ)|ξ |2 dVΣ < 2Ï â«Î£|Î¾ |2 dVΣ.Then#{eigenvalues of LËÎ£ that are strictly smaller than Ï} â¥ q2d .And the corresponding index estimates:Corollary 2.3.6. Let (M,g) be a 3-dimensional Riemannian manifold without bound-ary isometrically embedded in Rd and Σ a closed orientable immersed CMC sur-face of genus g in M and HΣ 6= 0. Suppose that every non-zero ξ âH 1(Σ) satisfiesâ«Î£2âi=1(|IIM(ei,Î¾ )|2+ |IIM(ei,?ξ )|2)âRM|Î¾ |2 dVΣ < â«Î£H2Î£|Î¾ |2 dVÎ£.12ThenIndex(Î£)â¥ gd.As mentioned above, the free boundary case is slightly different since, a priori,only harmonic vector fields that are tangential along the boundary provide admissi-ble test functions. However, it is still possible to find tangential vector fields whosedual vectors, despite not being tangential any more, have zero average along theboundary. This can be done as long as the dimension of the space of tangentialharmonic vector fields is sufficiently large. As a consequence the index estimatesare weaker.The respective results for free boundary CMC surfaces are:Theorem 2.3.8. Let (M,âM,g) be a 3-dimensional Riemannian manifold withboundary isometrically embedded in Rd and Σ a compact orientable immersedfree boundary CMC surface in M with HΣ 6= 0. Assume there exist a real numberÏ and a q-dimensional vector space Wq of harmonic vector fields on Σ that aretangential on the boundary âÎ£, such that any non-zero ξ âWq satisfiesâ«Î£2âi=1|IIM(ei,Î¾ )|2+ |IIM(ei,?ξ )|2 dVÎ£ââ«Î£(RM +H2Σ)|ξ |2 dVÎ£â2â«âÎ£HâM|Î¾ |2 dVâÎ£ < 2Ïâ«Î£|Î¾ |2 dVΣ.Then#{eigenvalues of LËÎ£ that are smaller than Ï} â¥ qâd2d .Corollary 2.3.9. Let (M,âM,g) be a 3-dimensional Riemannian manifold withboundary isometrically embedded inRd and Σ a compact orientable immersed freeboundary CMC surface in M with genus g, r boundary components and HΣ 6= 0.Suppose that every non-zero ξ âH 1T (Î£,âÎ£) satisfiesâ«Î£2âi=1|IIM(ei,Î¾ )|2+ |IIM(ei,?ξ )|2 dVÎ£ââ«Î£RM|Î¾ |2 dVÎ£â2â«âÎ£HâM|Î¾ |2 dVâÎ£ <â«Î£H2Î£|Î¾ | dVÎ£.13ThenIndex(Î£)â¥ 2g+ râ1âd2d.In the theorems above the operator LËÎ£ corresponds to the twisted Dirichleteigenvalue problem for the Jacobi operator on admissible functions [10]. The num-ber of negative eigenvalues is the index of Σ in the CMC sense.Let us mention that Cavalcande-de Oliveira in [15, 16] have obtained similarestimates independent of the inequality condition on the embedding of the ambi-ent manifold for the particular case of closed CMC surfaces in R3, S3 and freeboundary CMC surfaces in mean convex domains of these two cases.A conjecture attributed to Schoen and Marques-Neves (see [20, 44]) whichsays that minimal surfaces in 3-manifolds of positive Ricci curvature have indexbounded below by a linear function of their genus. In [7] Ambrozio-Carlotto-Sharphave confirmed this conjecture for a large class of ambient spaces. One may ask asimilar question for closed CMC surfaces. In this case, it follows from our resultsthat any closed CMC surface of sufficiently large mean curvature, depending onlyon the ambient space, in any closed 3-manifold has index bounded below by alinear function of the genus. In some examples we compute explicitly the lowerbound necessary for the mean curvature.As another application, in some examples we are able to prove that closedstable CMC surfaces of sufficiently large mean curvature have to be spheres. Inparticular, in T 2ÃR and in T 3 this partially supports a conjecture of Hauswirth-Perez-Romon-Ros regarding the isoperimetric profile of such ambient spaces. Tobe more specific, the authors conjecture that the isoperimetric profile is given byspheres, cylinders and pairs of planes. Since isoperimetric surfaces are stable CMCsurfaces, our result supports the section of the profile that is given by spheres.This chapter is divided as follows. In section 2 we cover some of the necessarybackground and notation used throughout the chapter. In section 3 we prove thatRosâ test functions are admissible and prove our main theorems. In section 4 wediscuss some applications and examples that satisfy the hypotheses of our mainresults.142.2 Preliminaries on CMC Surfaces and Morse IndexIn this section, we will give an overview some background material and establishnotation that will be used throughout the chapter.Let (M,âM,g) is a complete Riemannian manifold of dimension n+ 1, withpossibly empty boundary and let (Î£k,âÎ£) be a compact immersed k-dimensionalsubmanifold in M with boundary âÎ£ in âM. In case âM is empty, we consider âÎ£to also be empty, in which case we say Σk is a closed (compact without boundary)immersed submanifold in M. When k = n, we say that Σn is two-sided if there isa global unit normal vector field N along Σ in M. Denote the connection on Mand Σ by â and âÎ£, respectively. The second fundamental form of Σ is defined byAΣ(X ,Y ) =âXY ââÎ£XY , its shape operator with respect to N is denoted SΣ(X), themean curvature vector is given by ~HΣ = trΣ(AΣ) and its mean curvature is HΣ =g(~HΣ,N).By Nashâs embedding theorem we may assume M to be isometrically em-bedded into some Euclidean space Rd for some integer d. If we denote the Eu-clidean connection by D, then the second fundamental form of M in Rd is given byIIM(X ,Y ) =DXY ââXY , for any two tangent vectors X ,Y on M. It follows that forany two tangent vectors X ,Y on Σ, the following orthogonal decomposition holdsDXY = âÎ£XY +AΣ(X ,Y )+ IIM(X ,Y ). (2.2.1)Denote by â[p] the Hodge Laplacian of Σ acting on p-forms and let âââ de-note the rough Laplacian on vector fields. By choosing a local geodesic frame{e1, · · · ,en} of Σ, they are defined byâ[p]w = (dδ +δd)w, âââX =ânâi=1âeiâeiXfor p-forms w and vector fields X on Σ, where d is the exterior differential and δis the codifferential. If Σ is closed, then w is harmonic if and only if dw = 0 andδw = 0.Given a 1-form w, its dual vector field w is defined by g(w,X) = w(X) forany vector X . Reversely, given a vector field X we denote its dual 1-form X [(Y ) =g(X ,Y ).15Let us now restrict ourselves to when Σ is an orientable surface. We may define?X = (?X [), where ? is the Hodge operator with respect to the metric on Σ. TheLaplacian acting on a vector field X may be defined as â[1]X = â[1]X [. If X is avector field on Σ, we have ?dX [ = divΣ ?X and δX [ =âdivÎ£X . Hence, when Σ isclosed, X is harmonic if and only if divΣX = divΣ ?X = 0.When Σ is a closed surface we denote the space of harmonic vector fields on ΣasH 1(Σ) = {ξ â TΣ : div(ξ ) = div(?ξ ) = 0 on Σ }.If Σ has genus g then dimH 1(Σ) = 2g.When Σ has nonempty boundary, a vector field being harmonic is not equiva-lent to div(ξ ) = div(?ξ ) = 0. Instead we shall consider the following space:H 1T (Î£,âÎ£) ={ξ â TΣ : div(ξ ) = div(?ξ ) = 0 on Σand ξ is tangent along âÎ£}.Any vector field in this space is harmonic and tangential along âÎ£. By the Hodgetheorem and Poincaré-Lefschetz duality, we know that H 1T (Î£,âÎ£) is isomorphicto H1(Î£,âÎ£;R). Hence dimH 1T (Î£,âÎ£) = 2g+ râ1, where g is the genus and r isthe number of boundary components of Σ. See for example [49, Lemma A.0.1] fora proof.It is well know that Weitzenbockâs formula relates the Hodge Laplacian andrough Laplacian of a vector field. That is, for a vector field ξ on Σ we haveâ[1]Î¾ = âââÎ¾ +RicΣ(ξ ), (2.2.2)where RicΣ(ξ ) is defined by RicΣ(ξ ,X) = g(RicΣ(ξ ),X) for every tangent vectorX .Now, let us define the Morse index of a CMC hypersurface. Let Σ be a two-sided hypersurface with boundary âÎ£ in (M,âM). The first variation formula ofarea with respect to a normal variation induced by a function u on Σ is given byddtâ£â£â£â£t=0|Î¦t(Î£)|=ânâ«Î£uHÎ£ dVÎ£+â«âÎ£g(Î· ,X) dVâÎ£, (2.2.3)16where η is the outward pointing unit normal vector field along âÎ£, X is the or-thogonal projection of uN onto âM and Φt is the variation induced by u. A freeboundary constant mean curvature hypersurface, henceforth denoted free bound-ary CMC hypersurface, is a critical point of the area functional with respect tovariations that preserve enclosed volume. Here the enclosed volume representsthe signed volume enclosed between the hypersurface Σ and Φt(Σ), see [48] fordetails. This means that u must satisfyâ«Î£ u dVΣ = 0, in which case we call it anadmissible variation. Note that Σ must intersect âM orthogonally along âÎ£, whichis called the free boundary property, i.e., η = âÎ½ where ν is the inward pointingnormal vector field along âM. In this chapter we only consider CMC surfaces withnon-zero mean curvature.The quadratic form associated to the second variation of area of a free boundaryCMC hypersurface with respect to u isQΣ(u,u) =â«Î£|âu|2â(RicM(N,N)+ |AΣ|2)u2 dVÎ£ââ«âÎ£hâM(N,N) dVâÎ£, (2.2.4)where RicM(N,N) is the Ricci curvature of M in the direction N, |AΣ|2 = trΣ(STΣSΣ)is the square norm of second fundamental form and hâM(X ,Y ) = g(âXY,Î½) de-notes the scalar second fundamental form of âM in M with respect to ν .By the divergence theorem we can also write (2.2.4) asQΣ(u,u) =ââ«Î£uâu+(RicM(N,N)+ |AΣ|2)u2 dVÎ£+â«âÎ£uâuâÎ·âhâM(N,N)u2 dVâÎ£.We denote by LΣ = â+RicM(N,N)+ |AΣ|2 the Jacobi operator of Σ, whereâu = divÎ£(âÎ£u).The Morse index of a free boundary CMC hypersurface is defined as the indexof the quadratic form QΣ restricted to the subspace of smooth functions u on Σ thatare admissible, that is, such thatâ«Î£ udVΣ = 0. When the index is zero, we say thatthe hypersurface is stable in the CMC sense.This notion of index coincides with the number of negative eigenvalues for theoperator LËÎ£u = LÎ£uâ 1|Î£|â«Î£LÎ£udVÎ£ acting on admissible functions on Σ (see [10,17Proposition 2.2]); that is, of the following eigenvalue problem:ï£±ï£´ï£´ï£´ï£´ï£´ï£²ï£´ï£´ï£´ï£´ï£´ï£³LËÎ£u+Î»u = 0, in Î£âuâÎ·âhâM(N,N)u = 0, on âÎ£â«Î£udVÎ£ = 0.(2.2.5)The spectrum of LËÎ£ consists of eigenvalues Î»Ë1 â¤ Î»Ë2 â¤ . . . corresponding to ad-missible eigenfunctions ÏË1, ÏË2, . . . in L2(Σ), see [10] for details. Furthermore, theeigenvalues Î»Ëi satisfy the min-max characterization with respect to the quadraticform QΣ when restricted to admissible functions. That is, λk = infQÎ£(u,u)â«Î£ u2dVΣ, wherethe infimum is taken over all smooth admissible functions that are orthogonal tothe first kâ1 eigenfunctions ÏË1, . . . , ÏËkâ1.In the special case where the boundaries âM and âÎ£ are both empty, the cor-responding variation formulas are obtained similarly by simply making the termsthat involve the boundary as 0.Similarly we can define the Morse index of a closed CMC hypersurface as thenumber of negative eigenvalues of the Jacobi operator with respect to admissiblevariations. Note that the Jacobi operators are the same in closed and boundary case,the only difference is the boundary condition in the eigenvalue problem.Remark 2.2.1. Since we are considering CMC surfaces Σ with nonzero meancurvature, Σ is automatically two-sided in the ambient manifold M because themean curvature vector defines a trivialization of the normal bundle. As such, weonly have the following three possible combinations of CMC surfaces and ambi-ent space: (1) Σ is orientable and M is orientable; (2) Σ is orientable and M isnon-orientable; (3) Σ is non-orientable and M is non-orientable. The idea in thischapter applies to the first two cases while it does not work in the third case, seeRemark 2.3.7 for a more detailed discussion.2.3 Calculations and Main ResultsSimilarly to [47] and [7, 8] we intend to use the coordinates of harmonic vectorfields as test functions to obtain our index estimates. However, we must first verify18that such functions are admissible in the CMC sense.The following is a general result in arbitrary codimension that follows from thedivergence theorem.Lemma 2.3.1. Suppose (Î£k,âÎ£,g) is a k-dimensional submanifold (kâ¥ 2) isomet-rically embedded in some Euclidean space Rd .â¢ If âÎ£= /0 and ξ is a harmonic vector field inH 1(Σ), thenâ«Î£ãV,Î¾ ã dVΣ = 0,where V is a constant vector inRd . In particular, if k= 2 and Σ is orientable,we also have â«Î£ãV,?Î¾ ã dVΣ = 0.â¢ If âÎ£ 6= /0 and ξ is a tangential harmonic vector field inH 1T (Î£,âÎ£), thenâ«Î£ãV,Î¾ ã dVΣ = 0.Proof. Denote by f (x) = ãV,xã, then its gradient is given by âÎ£ f =V T , where V Tdenotes the projection of V onto Σ. If Σ is a closed submanifold and ξ âH 1(Σ)we get0 =â«Î£divÎ£( fξ ) =â«Î£ãâÎ£ f ,ξ ã+ f div(ξ ) =â«Î£ãâÎ£ f ,ξ ã=â«Î£ãV,Î¾ ã.When k = 2, the argument follows by substituting ?ξ and observing that it is alsoa harmonic vector field.In the case where Σ has nonempty boundary and ξ âH 1T (Σ) we getâ«Î£ãV,Î¾ ã=â«Î£divÎ£( fξ ) =â«âÎ£f ãÎ¾ ,Î·ã= 0The last equality is due to the tangential property of ξ . This completes the proofof the Lemma.Remark 2.3.2. When an orientable surface Σ has nonempty boundary and dimen-sion 2 this argument does not work for ?ξ since it may not be tangential on the19boundary when ξ is tangential. In fact, if ξ is tangential along the boundary then?ξ is necessarily orthogonal to the boundary. However, we may still be able to findtangential harmonic vector fields ξ such that f ãÎ¾ ,Î·ã has zero average along theboundary so its coordinates may still be admissible test functions.The following proposition involves calculations of coordinates of a harmonicvector field as test functions for the second variation formula both in closed caseand boundary case. In fact, the calculations in the closed case directly follow fromthe result in the boundary case.Proposition 2.3.3. Let (M,âM,g) be a 3-dimensional Riemannian manifold iso-metrically embedded in some Euclidean space Rd . Let (Î£,âÎ£) be an immersedsurface in M with free boundary âÎ£ in âM.â¢ Given a vector field ξ âH 1T (Î£,âÎ£), denoteu j = ãÎ¾ ,E jãwhere {E j}dj=1 is the canonical basis of Rd . Thendâj=1QÎ£(u j,u j) =â«Î£2âi=1|IIM(ei,Î¾ )|2+ |AΣ(ei,ξ )|2 dVÎ£ââ«Î£( |AΣ|22+RM2+H2Σ2)|ξ |2 dVÎ£ââ«âÎ£HâM|Î¾ |2 dVâÎ£(2.3.1)where {ei}2i=1 is an orthonormal frame on Σ, RM is the scalar curvature of Mand HâM denotes the mean curvature of âM with respect to the inner normalvector ν .â¢ In particular, if âM and âÎ£ are empty, for any vector field ξ âH 1(Σ), we20havedâj=1QÎ£(u j,u j) =â«Î£2âi=1|IIM(ei,Î¾ )|2+ |AΣ(ei,ξ )|2 dVÎ£ââ«Î£( |AΣ|22+RM2+H2Σ2)|ξ |2 dVΣ(2.3.2)Proof. Using a local orthonormal basis {ei}2i=1 on Σ we haveâÎ£u j =2âi=1eiãÎ¾ ,E jãei =2âi=1ãDeiÎ¾ ,E jãei.Then (2.2.1) givesDeiξ = âÎ£eiÎ¾ +AΣ(ei,ξ )+ IIM(ei,ξ ).Thus,dâj=1|âÎ£u j|2 =dâj=12âi=1|ãDeiÎ¾ ,E jã|2 (2.3.3)=2âi=1dâj=1ãâÎ£eiÎ¾ ,E jã2+ ãAÎ£(ei,Î¾ ),E jã2+ ãIIM(ei,Î¾ ),E jã2= |âÎ£Î¾ |2+2âi=1|AÎ£(ei,Î¾ )|2+ |IIM(ei,ξ )|2. (2.3.4)Gaussâ equation for Σ in M gives us2KΣ = RMâ2RicM(N,N)â|AÎ£|2+H2Î£.Hence,â«Î£RicM(N,N)u2j dVΣ =â«Î£(RM2â |AÎ£|22+H2Î£2âKÎ£)u2j dVΣ. (2.3.5)It follows from Weitzenbockâs formula (2.2.2), ξ being a harmonic vector field21and Σ being a surface thatâââÎ¾ =âKÎ£Î¾ .By computing the exterior derivative along âÎ£ we havedξ [(η ,ξ ) = ãâÎ£Î·Î¾ ,ξ ãâãâÎ£Î¾Î¾ ,Î·ã.Since dξ [ = 0,ãâÎ£Î·Î¾ ,ξ ã= ãâÎ£Î¾Î¾ ,Î·ã.Then it follows from the divergence theorem thatâ«Î£â|Î¾ |2 dVΣ =â«âÎ£âÎ£Î· |ξ |2 dVâÎ£= 2â«âÎ£ãâÎ£Î·Î¾ ,ξ ã dVâÎ£= 2â«âÎ£ãâÎ£Î¾Î¾ ,Î·ã dVâÎ£= â2â«âÎ£ãâÎ¾Î¾ ,Î½ã dVâÎ£= â2â«âÎ£hâM(Î¾ ,ξ ) dVâÎ£, (2.3.6)which together withâ|Î¾ |2 =â2ãâââÎ¾ ,ξ ã+2|âÎ£Î¾ |2implies thatâ«Î£|âÎ£Î¾ |2 dVΣ =â«Î£â|Î¾ |22dVÎ£+â«Î£ãâââÎ¾ ,ξ ã dVΣ= ââ«âÎ£hâM(Î¾ ,ξ ) dVâÎ£ââ«Î£KÎ£|Î¾ |2 dVΣ. (2.3.7)Note thatHâM|Î¾ |2 = hâM(Î¾ ,ξ )+hâM(N,N)|Î¾ |2. (2.3.8)Then (2.3.9) in the proposition follows from (2.3.4), (2.3.5), (2.3.7), (2.3.8) and(2.2.4). The formula (2.3.10) directly follows by ignoring boundary terms.22Similarly, for the dual harmonic vector field we haveProposition 2.3.4. Let (M,âM,g) be a 3-dimensional Riemannian manifold iso-metrically embedded in some Euclidean space Rd . Let (Î£,âÎ£) be an orientableimmersed surface in M with free boundary âÎ£ in âM.â¢ Given a vector field ξ âH 1T (Î£,âÎ£), denoteu?j = ã?Î¾ ,E jãwhere {E j}dj=1 is the canonical basis of Rd . Thendâj=1QÎ£(u?j ,u?j) =â«Î£2âi=1|IIM(ei,?Î¾ )|2+ |AΣ(ei,?ξ )|2 dVÎ£ââ«Î£( |AΣ|22+RM2+H2Σ2)|?ξ |2 dVÎ£ââ«âÎ£HâM|?Î¾ |2 dVâÎ£(2.3.9)where {ei}2i=1 is an orthonormal frame on Σ, RM is the scalar curvature of Mand HâM denotes the mean curvature of âM with respect to the inner normalvector ν .â¢ In particular, if âM and âÎ£ are empty, for any vector field ξ âH 1(Σ), wehavedâj=1QÎ£(u?j ,u?j) =â«Î£2âi=1|IIM(ei,?Î¾ )|2+ |AΣ(ei,?ξ )|2 dVÎ£ââ«Î£( |AΣ|22+RM2+H2Σ2)|?ξ |2 dVΣ(2.3.10)Proof. When Σ is orientable we have |?ξ |= |ξ | so the same result follows fromâ«Î£â|?Î¾ |2 =â«Î£â|Î¾ |2 dVΣ =â«âÎ£hâM(Î¾ ,ξ ) dVâÎ£. (2.3.11)This means that we have the same equation as (2.3.9) and (2.3.10) for ?ξ whetheror not its coordinate functions are admissible.23The following is the main theorem for closed CMC surfaces in 3-dimensionalmanifolds embedded in Euclidean space. It estimates the number of eigenvalues ofthe LËÎ£ below a certain threshold Ï under certain conditions on the embedding andthe geometry of M. This result is often referred to as a âconcentration of spectruminequalityâ (see [7]). Recall that LËÎ£u = LÎ£uââ«Î£LÎ£udVÎ£.Theorem 2.3.5. Let (M,g) be a 3-dimensional manifold without boundary isomet-rically embedded in Rd and Σ a closed orientable immersed CMC surface in Mwith HΣ 6= 0. Assume there exist a real number Ï and a q-dimensional vector spaceVq of harmonic vector fields on Σ such that any non-zero ξ â Vq satisfiesâ«Î£2âi=1(|IIM(ei,Î¾ )|2+ |IIM(ei,?ξ )|2)â (RM +H2Σ)|ξ |2 dVΣ < 2Ï â«Î£|Î¾ |2 dVΣ.Then#{eigenvalues of LËÎ£ that are strictly smaller than Ï} â¥ q2d .Proof. First, from Lemma 2.3.1 we have that u j and u?j = ã?Î¾ ,E jã for j = 1, · · · ,dare admissible test functions. Applying the same calculation for (2.3.10) to u?j wehave thatdâj=1QÎ£(u?j ,u?j) =â«Î£2âi=1|IIM(ei,?Î¾ )|2+ |AΣ(ei,?ξ )|2 dVÎ£ââ«Î£( |AΣ|22+RM2+H2Σ2)|?ξ |2 dVΣ.(2.3.12)Now, whenever ξ 6= 0 we may pick e1 = ξ|ξ | ,e2 = ?ξ|?ξ | as an orthonormal basis.Recalling that |ξ |= |?ξ |, we have at every point2âi=1|AÎ£(ei,Î¾ )|2+ |AΣ(ei,?ξ )|2 = âi, j=1,2|AΣ(ei,e j)|2|ξ |2 = |AΣ|2|ξ |2. (2.3.13)24From this we obtain, by adding (2.3.10) and (2.3.12),dâj=1QÎ£(u j,u j)+QΣ(u?j ,u?j) =â«Î£2âi=1|IIM(ei,Î¾ )|2+ |IIM(ei,?ξ )|2 dVÎ£ââ«Î£(RM +H2Σ)|ξ |2 dVΣ.For convenience, let k = #{ eigenvalues of LËÎ£ that are strictly smaller than Ï}and let ÏË1, · · · , ÏËk be the eigenfunctions of the operator LËÎ£ corresponding to theeigenvalues Î»Ë1 â¤ Â·Â· · â¤ Î»Ëk < Ï . Consider following the linear map defined byF : Vq ââ R2dkξ 7ââ [â«Î£u jÏËÎ± dVÎ£,â«Î£u?j ÏËÎ± dVΣ],where α = 1, · · · ,k and j = 1, · · · ,d. By the Rank-Nullity theorem, if 2dk < q,then there exists a nonzero harmonic vector field ξ such that u j,u?j are orthogonalto the first k eigenfunctions of LËÎ£. From the Min-max principle for the operator LËÎ£it follows thatdâj=1QÎ£(u j,u j)+QΣ(u?j ,u?j)â¥ 2Î»k+1â«Î£|Î¾ |2 dVΣ â¥ 2Ïâ«Î£|Î¾ |2 dVΣwhich contradicts the assumption in the proposition. Thus 2dk â¥ q, or k â¥ q2d asclaimed.We remark that the same proof works to estimate the number of eigenvaluesof the Jacobi operator. However, such an estimate would not directly give indexestimates in the CMC sense.As a corollary, letting Ï = 0 we can get an estimate on numbers of negativeeigenvalues of LËÎ£, that is, an index estimate.Corollary 2.3.6. Let (M,g) be a 3-dimensional Riemannian manifold without bound-ary isometrically embedded in Rd and Σ a closed orientable immersed CMC sur-25face of genus g in M with HΣ 6= 0. Suppose that every non-zero ξ âH 1(Σ) satisfiesâ«Î£2âi=1(|IIM(ei,Î¾ )|2+ |IIM(ei,?ξ )|2)âRM|Î¾ |2 dVΣ < â«Î£H2Î£|Î¾ |2 dVÎ£.Then,Index(Î£)â¥ gdRemark 2.3.7. In Theorem 2.3.5 and Corollary 2.3.6, the orientability assumptionis essential for our method. Since in this case ?ξ is a globally well-defined vectorfield. When Σ is non-orientable, in which case M has to be non-orientable, wecould consider the oriented double covering Î£Ë of Σ which would be a two-sidedCMC immersed surface in M. The Hodge star operator ? on Î£Ë would be welldefined but given a harmonic vector field ξ invariant under the change of sheets,the corresponding ?ξ would be anti-invariant, or vice-versa. That is, only one ofthese can define a valid test function of Σ. However, our method requires both testfunctions to eliminate certain undesired terms involving the second fundamentalform of Σ when computing the second variation in that direction.The next result is the corresponding theorem for free boundary CMC surfacesin 3-dimensional manifolds with boundary. From Remark 2.3.2 we observe u?jmay not be admissible, however, it is possible to find suitable vector fields whenthe dimension ofH 1T (Î£,âÎ£) is sufficiently large.Theorem 2.3.8. Let (M,âM,g) be a 3-dimensional Riemannian manifold withboundary isometrically embedded in Rd and Σ a compact orientable immersedfree boundary CMC surface in M with HΣ 6= 0. Assume there exist a real numberÏ and a q-dimensional vector space Wq of harmonic vector fields on Σ that aretangential to âÎ£ along the boundary âÎ£, such that any non-zero ξ âWq satisfiesâ«Î£2âi=1|IIM(ei,Î¾ )|2+ |IIM(ei,?ξ )|2 dVÎ£ââ«Î£(RM +H2Σ)|ξ |2 dVÎ£â2â«âÎ£HâM|Î¾ |2 dVâÎ£ < 2Ïâ«Î£|Î¾ |2 dVΣ.Then#{eigenvalues of LËÎ£ that are smaller than Ï} â¥ qâd2d .26Proof. From Proposition 2.3.4 we know thatdâj=1QÎ£(u?j ,u?j) =â«Î£2âi=1|IIM(ei,?Î¾ )|2+ |AΣ(ei,?ξ )|2 dVÎ£ââ«Î£( |AΣ|22+RM2+H2Σ2)|ξ |2 dVÎ£ââ«âÎ£HâM|Î¾ |2 dVâÎ£.(2.3.14)Summing up (2.3.14) and (2.3.9), together with (2.3.13) givesdâj=1QÎ£(u j,u j)+QΣ(u?j ,u?j) =â«Î£2âi=1|IIM(ei,Î¾ )|2+ |IIM(ei,?ξ )|2 dVÎ£ââ«Î£(RM +H2Σ)|ξ |2 dVÎ£â2â«âÎ£HâM|Î¾ |2 dVâÎ£.Let k= #{eigenvalues of LËÎ£ that are smaller than Ï} and let ÏË1, · · · , ÏËk be the eigen-functions of LËÎ£ corresponding to the eigenvalues Î»Ë1 â¤ Â·Â· · â¤ Î»Ëk < Ï . Consider thelinear map defined byF : Wq ââ R2dk+dξ 7ââ [â«Î£u jÏÎ± dVÎ£,â«Î£u?jÏÎ± dVÎ£,â«Î£u?j dVΣ],where α = 1, · · · ,k and j = 1, · · · ,d. By the Rank-Nullity theorem, if 2dk+d < q,then there exists a nonzero harmonic tangential vector field ξ such that u j,u?j areorthogonal to the first k eigenfunctions of LËÎ£. From the Min-max principle for LËÎ£it follows thatdâj=1QÎ£(u j,u j)+QΣ(u?j ,u?j)â¥ 2Î»k+1â«Î£|Î¾ |2 dVΣ â¥ 2Ïâ«Î£|Î¾ |2 dVΣwhich contradicts the assumption in the proposition. Thus 2dk+d â¥ q, or kâ¥ qâd2das claimed.Correspondingly, we have an index estimate for free boundary CMC surfaces.27Corollary 2.3.9. Let (M,âM,g) be a 3-dimensional Riemannian manifold withboundary isometrically embedded inRd and Σ a compact orientable immersed freeboundary CMC surface in M with genus g, r boundary components and HΣ 6= 0.Suppose that every non-zero ξ âH 1T (Î£,âÎ£) satisfiesâ«Î£2âi=1|IIM(ei,Î¾ )|2+ |IIM(ei,?ξ )|2 dVÎ£ââ«Î£RM|Î¾ |2 dVÎ£â2â«âÎ£HâM|Î¾ |2 dVâÎ£ <â«Î£H2Î£|Î¾ | dVÎ£.ThenIndex(Î£)â¥ 2g+ râ1âd2d.2.4 Applications in the Classification of Stable CMCSurfaces and Isoperimetric ProblemsIn this section we discuss some examples for which our index estimates for CMCsurfaces apply. For some special manifolds that can be canonically embedded inRn we characterize their stable CMC surfaces for a range of values of the meancurvature.Let us first discuss some applications in the case of closed CMC surfaces in aRiemannian manifold without boundary.Theorem 2.4.1. Let M be a Riemannian manifold of dimension 3 immersed inRd with second fundamental form IIM and scalar curvature RM. If Σ is a closedorientable immersed CMC surface with mean curvature H2Σ > sup |IIM|2â infRMand HΣ 6= 0 thenIndex(Î£)â¥ gd.In particular, if Σ is stable then it must be a sphere.Proof. Whenever ξ 6= 0 we may pick e1 = ξ|ξ | ,e2 = ?ξ|?ξ | as an orthonormal basis.Recalling that |ξ |= |?ξ |, we have2âi=1|IIM(ei,Î¾ )|2+ |IIM(ei,?ξ )|2 = âi, j=1,2|IIM(ei,e j)|2|ξ |228at every point. Since âi, j=1,2 |IIM(ei,e j)|2 â¤ |IIM|2 we obtainâ«Î£2âi=1(|IIM(ei,Î¾ )|2+ |IIM(ei,?ξ )|2)âRM|Î¾ |2 dVΣ â¤(sup |IIM|2â infRM)â«Î£|Î¾ |2dVΣ.As long as H2Σ > sup |IIM|2âinfRM the condition of Corollary 2.3.6 is satisfied.The following is a direct application of the Nash embedding theorem and thetheorem above.Corollary 2.4.2. Let M be a closed Riemannian manifold of dimension 3. Thereexist constants C > 0 and H0 â¥ 0 depending on M such that for every closed ori-entable immersed CMC surface Σ of genus g in M and mean curvature |HΣ|> H0,Index(Î£)â¥CgIn particular, if Σ is stable then it must be a sphere.Remark 2.4.3. The result above proves a CMC version of a conjecture of Schoenand Marques-Neves for CMC surfaces of sufficiently large mean curvature in arbi-trary 3-manifolds.First we would like to mention that Barbosa-do Carmo-Eschenburg [12] havestudied CMC hypersurfaces in simply connected spaces of constant sectional cur-vature and proved that geodesic spheres are the only stable ones. In [16] Cavalcante-de Oliveira proved this using similar index estimates. Let us now present some ex-amples that satisfy the conditions of the above results using canonical embeddings.Closed CMC surfaces in S2ÃR.In [54] Souam proved that every stable CMC in S2ÃR is a rotational sphere.We are going to give an alternative proof using our index estimates.29Theorem 2.4.4 ([54, 4.1]). Let Σ be a closed immersed CMC surface in S2(r)ÃR.Then,Index(Î£)â¥ g4.In particular, if in addition Σ is stable, then it is a sphere.Proof. Consider the canonical embedding of S2(r)ÃR in R4. Its second funda-mental form is given by IIS2(r)ÃR(X ,Y ) = 1r ãpiâX ,piâY ã, where pi denotes the pro-jection onto S2(r). In particular, |IIS2(r)ÃR|2 = 2r2 . Since RS2(r)ÃR = 2r2 , it followsfrom Theorem 2.4.1 that Index(Σ) â¥ g4 whenever HΣ 6= 0. It remains to considerthe case HΣ = 0. However, by moving spherical slices, it follows from the maxi-mum principle that the only closed minimal surfaces in are slices S2(r)Ã{t} thusfinishing the proof.Closed CMC surfaces in T 2(α,β )ÃR.Let T 2(α,β ) denote the 2-torus defined by the quotient of R2 with respect tothe lattice Î(Î±,Î² ) generated by {(0,1),(α,β )} satisfying 0 â¤ α â¤ 12 , β > 0 andα2 + β 2 â¥ 1. We remark that any flat torus is isometric, up to dilations, to theabove.Let us consider the following map f : R2 â R6, f = ( f1, f2, f3, f4, f5, f6) de-fined byf1 =C1 cos(a1u+b1v)f2 =C1 sin(a1u+b1v)f3 =C2 cos(a2u+b2v)f4 =C2 sin(a2u+b2v)f5 =C3 cos(b3v)f6 =C3 sin(b3v).If we choose ai = 2piki, bi = 2piâÎ±2+Î² 2(liâ ki Î±Î²âÎ±2+Î² 2)for positive integerski, li, i=1,2 and b3 = 2piâÎ±2+Î² 2 then f becomes an embedding of T2(α,β ) into R6.30Furthermore, as long as li,ki satisfyÎ±Î²âÎ±2+Î² 2 <l1k1<(Î±Î²+1)âÎ±2+Î² 2(Î±Î²â1)âÎ±2+Î² 2 <l1k1<Î±Î²âÎ±2+Î² 2,it is always possible to find positive constants C1,C2,C3 so that f is an isometricembedding (see [41]).Theorem 2.4.5. Let ki, li, i = 1,2 be positive integers and C1,C2,C3 > 0 so thatf defined above is an isometric embedding of T 2(α,β ) into R6. If Σ is closedimmersed CMC surface of genus g in T 2(α,β )ÃR with mean curvature H2Σ >C21(a21+b21)2+C22(a22+b22)2+C23b43, thenIndex(Î£)â¥ g7.In particular, if in addition Σ is stable, then it is a sphere.Proof. We embed T 2(α,β )ÃR in R7 using f Ã id and choose the following nor-mal frame:N1 = (âcos(a1u+b1v),âsin(a1u+b1v),0,0,0,0,0)N2 = (0,0,âcos(a2u+b2v),âsin(a2u+b2v),0,0,0)N3 = (0,0,0,0,âcos(b3v),âsin(b3v),0)N4 = M(a2C2 sin(a1u+b1v),âa2C2 cos(a1u+b1v),âa1C1 sin(a2u+b2v),a1C1 cos(a2u+b2v),â(a2b1âa1b2)C1C2b3C3 sin(b3v),(a2b1âa1b2)C1C2b3C3cos(b3v),0)with M = 1a21C21+a22C22+C21C22b23C23(a2b1âa1b2)2.31The shape operators for each normal direction and basis â fâu ,â fâv are given byS1 =C1(a21 a1b1a1b1 b21),S2 =C2(a22 a2b2a2b2 b22),S3 =C3(0 00 b23),S4 = 0.So, we can compute |S1|2 =C21(a21 + b21)2, |S2|2 =C22(a22 + b22)2 and |S3|2 =C23b43.From this it follows that|IIT 2ÃR|2 =C21(a21+b21)2+C22(a22+b22)2+C23b43.Since RT 2ÃR = 0, the result follows from Theorem 2.4.1.In [37] Hauswirth-Perez-Romon-Ros study doubly periodic isoperimetric sur-faces and conjecture that the isoperimetric profile of T 2(α,β )ÃR is given byspheres, cylinders around lines and pairs of planes. The authors then prove theconjecture for sufficiently large values of β . The result above provides supportfor the spherical part of the isoperimetric profile conjecture for any given valuesof α,β . It also says that any genus 1 isoperimetric surface, hence stable CMC,has a computable upper bound on the mean curvature, thus placing it further in theisoperimetric profile.We would like to highlight two special cases. First, the hexagonal torus T 2(12 ,â32 )which is a critical case in Hauswirth-Perez-Romon-Rosâ proof. In this case we maychoose, for example, a1 = b3 = 2pi , a2 = 4pi , b1 = 2pi(1â 1â3) and b2 = 2pi(1â2â3).Then, to make f an isometric embedding we solve an underdetermined linear sys-tem so we may pick C21 = 1, C22 =12(â3â 1) and C23 = 11+2â36 . However, theseare not unique choices and are far from being optimal in the sense that the lowerbound for HΣ on the result above is minimal.Secondly, the case of α = 0, that is, rectangular tori. Rectangular tori can beembedded in R4 canonically and the estimate on HΣ is better.Corollary 2.4.6. Let T 2 = T 2(0,β ), β â¥ 1, be a rectangular torus. If Σ is aclosed immersed CMC surface of genus g in T 2ÃR with mean curvature H2Σ >(2pi)2(1+ 1β 2), thenIndex(Î£)â¥ g5.32In particular, if in addition Σ is stable, then it is a sphere.Proof. In the case of a rectangular tori the embedding reduces tof1 =12picos(2piu)f2 =12pisin(2piu)f3 =β2picos(2piβv)f4 =β2pisin(2piβv).We then embed T 2 ÃR in R5 and using the canonical normal frame the shapeoperators satisfy |S1|2 = (2pi)2 and |S2|2 =(2piβ)2. From this it follows that|IIT 2ÃR|2 = (2pi)2(1+1β 2).Since the scalar curvature is zero, the result follows from Theorem 2.4.1.Closed CMC surfaces in T 3.Let us discuss the case of triply periodic CMC surfaces in R3. Similarly tothe doubly periodic case, Hauswirth-Perez-Romon-Ros conjecture that the isoperi-metric profile on T 3 is given only by spheres, cylinders around lines and pairs ofplanes. The following result supports the conjecture on the spherical branch inthe same manner as above. Any rectangular torus is equivalent, up to dilations, tothe quotient of R3 by the lattice generated by { (1,0,0), (0,A,0), (0,0,B) }. Weidentify such tori with S1(1)ÃS1(r1)ÃS1(r2) with r1 = 2piA and r2 = 2piB .Theorem 2.4.7. Let T 3 = S1(1)Ã S1(r1)Ã S1(r2) be a rectangular 3-torus. If Σis a closed immersed CMC surface of genus g in T 3 with mean curvature H2Σ >1+ 1r21+ 1r22, thenIndex(Î£)â¥ g6.In particular, if in addition Σ is stable, then it is a sphere.33Proof. Consider the canonical isometric embedding of T 3 into R6. Using the usualnormal frame the corresponding shape operators satisfy |S1|2 = 1, |S2|2 = 1r21 and|S3|2 = 1r22 . Thus,|IIM|2 = 1+ 1r21+1r22.Since the ambient space is flat, the statement follows from Theorem 2.4.1.Closed CMC surfaces in Berger Spheres S3b(Îº,Ï)Let S3 â C2 be the unit sphere. We denote by S3b(Îº,Ï) the Berger Spherewith metric gÎº,Ï(X ,Y ) = 4κ(g(X ,Y +(4Ï2Îº â1)g(X ,V )g(Y,V ))), where g is theEuclidean metric and V = (iz, iw) is a Killing vector field on S3b(Îº,Ï).Following Torralbo-Urbano [56, Section 2.1], as long as κ â 4Ï2 > 0 the au-thors construct an isometric embedding of S3b(Îº,Ï) into the complex projectivespace CP2(κ â 4Ï2) as a geodesic sphere. The space CP2(κ â 4Ï2) is the usualcomplex projective space endowed with the a dilation of the Fubini-Study metric4Îºâ4Ï2 gFS with constant holomorphic sectional curvature Îºâ4Ï2.They then follow to embedCP2(Îºâ4Ï2) intoR8 and obtain a stability criterionsimilar to our index estimates in the case when the CMC surface is stable (see[56, Lemma 6.4]). By using the calculations already carried out in their paper weare able to improve their stability criterion in order to obtain the following indexestimates.Theorem 2.4.8. Let S3b(Îº,Ï) be a Berger Sphere with κ â 4Ï2 > 0. If Σ is aclosed immersed CMC surface of genus g in S3b(Îº,Ï) with mean curvature H2Σ >Ï2( Îº4Ï2 â3)( Îº4Ï2 +1)and HΣ 6= 0, thenIndex(Î£)â¥ g8.In particular, if Îº4Ï2 â (1,3) then the index estimate is valid for all values of HΣ 6= 0.Proof. For short we write S3b = S3b(Îº,Ï). Consider the isometric embeddings S3b âCP2(Îº â 4Ï2) â R8 constructed in [56, Section 2.1] and denote by IIS3b second34fundamental form of S3b in R8. In the proof of [56, Theorem 6.5:Case(4)] Torralbo-Urbano compute the following:2âi, j=1|IIS3b(ei,e j)|2 =â6Ï2+2Îº+ Ï2( Îº4Ï2â1)2(1âC2)2,where e1,e2 is an orthonormal frame on Σ and C = Îº4Ï gÎº,Ï(N,V ) satisfies 0â¤C2 â¤1.Since RS3b = 2(κ â Ï2), it follows from the proof of Theorem 2.4.1 that theindex estimate is valid as long as H2Σ > Ï2( Îº4Ï2 â1)2â 4Ï2, which concludes theproof.Closed CMC surfaces in pinched manifolds.Similar to [7] we would like to mention that the following examples remainvalid in the CMC case.Theorem 2.4.9. Let M be an immersed manifold in Rd with RM > C|~HM|2 forsome C > 0. If Σ is a closed immersed orientable CMC surface of genus g in Mwith mean curvature H2Σ â¥ |~HM|2(1â2C) and HΣ 6= 0, thenIndex(Î£)â¥ gd.In particular, if RM > 12 |~HM|2 then any CMC surface satisfies the index estimateabove.Proof. Observe that Gaussâ equation on M implies that |IIM|2 = |~HM|2âRM. As aconsequence, we havesup |IIM|2â infRM < |~HM|2(1â2C).The result follows from Theorem 2.4.1.In the case of closed hypersurfaces of R4 we have the following weaker pinch-ing result. Compare to [7, Remark 5.3].35Theorem 2.4.10. Let M be an embedded convex two-sided hypersurface of R4with principal curvatures 0 â¥ k1 â¥ k2 â¥ k3 with respect to the outward normalvector field. Suppose M satisfies a pinching condition k3k1 < C, for some C > 0.If Σ is a closed immersed CMC surface of genus g in M with mean curvatureH2Σ â¥ 3k21(C2â2) and HΣ 6= 0, thenIndex(Î£)â¥ g4.In particular, if k3k1 <â2 then any CMC surface satisfies the index estimate above.Proof. Firstly observe that |IIM|2 = k21 + k22 + k23 â¤ 3k23. Secondly, it follows fromGaussâ equation that RM = 2(k1k2+ k1k3+ k2k3)â¥ 6k21. That is,sup |IIM|2â infRM â¤ 3k21(k23k21â2)< 3k21(C2â2).The result follows from Theorem 2.4.1.Free boundary CMC surfacesTo avoid repetition, let us mention that a similar index estimate is valid forfree boundary CMC surfaces in a mean convex domain of any of the above exam-ples. More especifically, let Σ be a free boundary CMC surface of genus g andr boundary components in a domain with mean convex boundary, with respect tothe inward normal direction, in any of the examples above. Suppose Σ satisfiesthe same mean curvature condition as in the closed case in its respective ambientmanifold. Then Index(Σ) â¥ 2g+râ1âd2d where d is the dimension of the Euclideanspace where the respective ambient manifold is embedded.Let us highlight the special case of mean convex domains in R3. The prooffollows directly from Corollary 2.3.9. This particular case was first proved byCavalcante-de Oliveira [15].Theorem 2.4.11 ([15, Theorem 1.1]). Let (M,âM) be a mean convex, with respectto the inward normal vector, domain of R3. If Σ is a free boundary orientable CMC36surface of M of genus g, r boundary components and HΣ 6= 0, thenIndex(Î£)â¥ 2g+ râ46.37Chapter 3Compactness of the Space of FreeBoundary CMC Surfaces withBounded Topology3.1 IntroductionA constant mean curvature (CMC) surface Σ is a critical point of the area functionalwith respect to variations that preserve enclosed volume. As a consequence of thefirst variation of area, the scalar mean curvature has to be constant for a givenchoice of normal direction. If Σ is an immersed surface with boundary immersedin a manifold N with boundary and with âÎ£ â âN, we say the surface is freeboundary CMC if it is a critical point with respect to variations that are in additiontangent along âN.Although most results about minimal surfaces have an equivalent version forCMC surfaces, there are many distinctions between their behaviour. For example,when the mean curvature is non-zero, the mean curvature vector defines a trivializa-tion of the normal bundle. That is, every CMC surface in a 3-manifold is 2-sided.On the other hand, CMC surfaces may have tangential self-touching points as longas the mean curvature vector points in opposite directions at those points.The goal of this chapter is to prove a CMC version of Fraser-Liâs result [26]38for free boundary minimal surfaces. We prove:Theorem 3.1.1. Let N be a compact 3-dimensional manifold with boundary. Sup-pose HâN â¥H0 and let Σi be a sequence of free boundary embedded CMC surfaceswith mean curvature Hi, genus gi and number of ends ri satisfying:(a) |Hi| â¤ H0;(b) gi â¤ g0;(c) ri â¤ r0;(d) area(Î£i)â¤ A0 and(e) length(âÎ£i)â¤ L0.Then there exists a smooth properly almost embedded CMC surface Σ â Nand a finite set Î â Σ such that, up to a subsequence, Σi converges to Σ locallygraphically in the Ck topology on compact sets of N \u00CE for all k â¥ 2. Moreover, ifΣ is minimal then it is properly embedded.If in addition (N,âN) satisfies either RicN > 0 and AâN â¥ 0 or RicN â¥ 0 andAâN > 0, then:(i) when HΣ = 0 the convergence is at most 2-sheeted;(ii) when HΣ 6= 0, then the convergence is 1-sheeted away from Î.Let us highlight the main differences that justify the extra hypotheses and theweaker compactness for embedded CMC surfaces.First, we mention that there are no Steklov eigenvalue estimates for free bound-ary CMC surfaces. Consequently, the hypothesis of length bound remains crucial.Secondly, there is no suitable isoperimetric inequality that allows us to remove thebound on the area. One could exchange the length bound condition by stability ofthe surface (see [43]) but as seen in [4], stable free boundary CMC surfaces havebounded topology, so this condition would be topologically restrictive. Thirdly,even under positive Ricci curvature of the ambient space, a sequence of free bound-ary CMC surfaces may have a neck-pinching phenomenum where the norm of the39second fundamental form blows-up at a point in the limit. Naturally the conver-gence is not smooth along these points where curvature is accumulating. Finally,the maximum principle for CMC surfaces only applies when their mean curvaturevectors point in the same direction. That is, a sequence of embedded free boundaryCMC surfaces may touch tangentially in the limit as long as the limiting surface isnot minimal.Despite the lack of certain useful technical results, the proof of the theoremrelies on the same main ideas, each of which is proved using a less optimal tech-nique to compensate for the above limitations. These are: L2-curvature boundsfrom the topological bound, improvement to local pointwise curvature estimates,a removable singularity theorem for interior and boundary points and constructionof positive eigenfunctions for the Jacobi operator to study convergence under acurvature condition on the ambient space.Let us briefly address our approach to each of the above tools. The integralcurvature bounds follow directly from the Gauss-Bonnet Theorem, from whichthe fact the mean curvature is non-zero introduces an extra term of area and thegeodesic curvature along the boundary brings in a term of boundary length. Thelocal pointwise curvature estimate comes from a blow-up argument as in [58] andSchauder estimates. It becomes relevant that the blow-up of a CMC surface arounda point is a minimal surface in R3. Unlike [26, Theorem 4.1] we do not knowwhether a CMC surface is conformally equivalent to a punctured Riemann surfaceso we do not have the branch point structure to prove the removable singularitytheorem. This will follow from a blow-up argument to prove that tangent cones arein fact planes (or half-planes) and a local free boundary CMC foliation argumentto prove that it is also unique. These are the same ideas as in [58] together with themethods in [9] to deal with boundary singularity points. In fact, most of the calcu-lations are done in the latter reference and only minor adaptations are necessary forthe non-minimal case. Finally, the construction of positive eigenfunctions for theJacobi operator on the limiting surface is a well known method (see [22, 51, 52])and again, most of the necessary calculations are done in [9].The extra condition on the mean curvature of âN is only necessary to applythe maximum principle for CMC surfaces and ensure that the interior of the limit-ing surface is properly immersed in N. Removing this condition would allow for40interior tangential touching points between âN and the limit surface.Finally, we mention that the corresponding result for closed CMC surfaces wasproved by Sun in [55]. Although the core ideas are the same, the approach tocertain steps is different and we focus on the behaviour at the boundary.This chapter is divided as follows. Section 2 establishes notations, necessarydefinitions and we prove the geometric version of Schauder estimates. In section 3we prove the local pointwise curvature estimates from uniform integral curvaturebounds. The Removable Singularity Theorem is proved in section 4 and we writethe proof for the specific case of a singularity along the boundary. Section 5 isdedicated to prove the Compactness Theorem.3.2 Preliminaries on Convergence of SurfacesIn this section we establish notation and preliminary results that will be usedthroughout this chapter.Let (N3,âN,g) be a 3-dimensional manifold with non-empty boundary and asmooth Riemannian metric g.Definition 3.2.1. We say that an immersed surface Î£âN with non-empty boundaryis properly almost embedded in an open set U â N if (Σ âÎ£)â©U is properlyimmersed in (N \u00E2N)â©U, âÎ£â âNâ©U and there exists a setS â Σ such that(a) ΣS is embedded;(b) for each p âS there exists a neighbourhood V of p in U such that Î£â©V isa union of connected components Wj, j = 1, . . . , lp, each Wj is embedded, foreach j 6= jâ² we have Wj lying to one side of Wjâ² and Wj â©Wjâ² âS â©V .The set S is called the self-touching set of Σ. We say Σ is free boundary if inaddition Σ is orthogonal to âN.The following lemma is a standard application of Schauder estimates in ge-ometry. It will allow us to improve from C1,α to C2,α graphical convergence ofsurfaces, given pointwise curvature estimates.Lemma 3.2.2. Let N be a three dimensional compact manifold, A > 0 and f âC0,Î±(N) with | f |0,α <H0. There exists r0(N,A)> 0 and C0(N,A,H0)> 0 such that41for any C2 immersed surface with free boundary Σ â N with HΣ = f and x0 â Σsatisfying|AΣ(x)| â¤ A on {x â Σ : dΣ(x,x0)< r0}there exists a function u defined on a subset of Tx0Σ with the following properties:(i) {x â Σ : dΣ(x,x0)< r0} is the graph of u over the exponential function;(ii) the domain of u contains a ball centered at 0 and radius r02 and(iii) âu|B r04(0)â2,Î± <C0.Proof. We may assume, that r0 > 0 is smaller than the convexity radius of N so that{x â Σ : dΣ(x,x0) < r0} is contained in a geodesic ball in N. Hence, without lossof generality we may assume that {x â Σ : dΣ(x,x0) < r0} is contained in a smallball in R3 with a metric g, x0 = 0 and Tx0Σ = R2Ã{0}. We may further assumethat g is equivalent to the Euclidean metric g0, that is, câ1g0 â¤ g â¤ cg0 where theconstant c > 0 depends only on the geometry of N.Claim 1. If r0 is sufficiently small, depending only on the geometry of N and A,then for all x â {x â Σ : dΣ(x,0) < r0} we have |Î½(x)â ν(0)|g0 â¤ CAdΣ(x,0) forsome C > 0 depending only on the geometry of N.Indeed, let γ â Σ be a curve parametrised by arc-length joining 0 to x. If wedenote by â0 the Euclidean connection, then it follows that|Î½(x)âÎ½(0)|g0 â¤ supt|â0Î³ËÎ½ |g0 l(γ)Since we are working in geodesic coordinates then â0Î³Ë = âÎ³Ë +O(|γ|), hence|â0Î³ËÎ½ |g0 â¤ c|â0Î³ËÎ½ |g â¤ c|âÎ³ËÎ½ |g+ |γ|C â¤ cA+ r0C,for some C > 0 depending only on the geometry of N. If we pick r0C < cA weobtain|Î½(x)âÎ½(0)|g0 â¤ 2cAl(γ).Taking the limit as the length of γ tends to dΣ(x,0) we conclude the claim as de-sired.42Now, suppose that CAr0 â¤ 1, then {x â Σ : dΣ(x,0) â¤ r0} is the graph ofa function u, otherwise we would have ν(x) perpendicular to ν(0) for some xhence |Î½(x)â ν(0)| > 1. Furthermore, it follows that sup |âu| â¤ 2CAr0 as longas C2A2r20 <34 .Let us denote byâ¦â T0Σ the domain of u and take δ > 0 the largest radius suchthat Bδ (0)ââ¦. In particular there exists y¯â âBÎ´ (0) such that dΣ((y¯,u(y¯)),(0,0))=r0.Claim 2. For any pair y1,y2 â Bδ (0) we have dΣ((y1,u(y1)),(y2,u(y2))) â¤ (1+(2CAr0)2)12 |y1â y2| and, if 4CAr0 < 1 then r02 â¤ δ .We have y1 + t(y2â y1) â Bδ (0) â â¦ for t â [0,1], from which follows thatdÎ£((y1,u(y1)),(y2,u(y2)))â¤â« 10 (1+|âu(y1+t(y2ây1))|2)12 |y1ây2|dt and it provesthe inequality.For the second part observe that if 4CAr0 < 1 then dΣ((y¯,u(y¯)),(0,0)) â¤ 32 |y¯|.Suppose that δ < r02 , then |y¯|< r02 and dÎ£((y¯,u(y¯)),0)â¤ 3r04 which is a contradictionand it proves the claim.Claim 3. If 4CAr0 < 1 then âu restricted to B r02(0) is a Lipschitz function ofLipschitz constant less than 32 A.The length of the second fundamental form is a multiple of norm of the Hessianof u, thus |AΣ| â¤A implies |Hess(u)| â¤ (1+ |âu|2) 12 A< 2A whenever 4CAr0 < 1. Itfollows that for any y1,y2 ââ¦, |âu(y1)ââu(y2)| â¤ 2AdΣ((y1,u(y1)),(y2,u(y2))).From the previous claim we get thatdÎ£((y1,u(y1)),(y2,u(y2)))â¤ (1+(2CAr0)2) 12 |y1â y2| â¤ 32 |y1â y2|,thus proving the claim.Finally, on B r02(0) the function u satisfies the equation div(âuâ1+|âu|2)= f .The coefficients of the equation have C0,α norm depending on the metric g and theLipschitz constant of âu. In particular all coefficients have C0,α norm dependingonly on N and A.Let us consider the case without boundary components on {x â Σ : dΣ(x,x0)<r0}. Then Schauder estimates [33, Corollary 6.3] implies that there exists a con-43stant C = C(N,A) such that âu|B r04(0)â2,Î± < C(âu|B r02(0)â0 + â fâ0,Î±). Note that|u(y)â u(0)| < 2Ar0|y| < 2Ar202 and u(0) = 0. That is, âu|B r04(0)â2,Î± <C0 for C0 =C0(N,A,H0).In case there are boundary components of Σ on {x â Σ : dΣ(x,x0) < r0} thenSchauder estimates ([2, Theorem 7.1]) gives us the same inequality with an ex-tra term depending on the C1,α norm of the inward normal derivative along theboundary. Since Σ is free boundary, u satisfies homogenous Neumann boundaryconditions in which case the extra term vanishes and we obtain the same result.In the following we make precise the notion of graphical convergence of sur-faces (see also [51]).Definition 3.2.3. Let N be a 3-manifold, with or without boundary, Σi a sequence ofsmooth surfaces in N and Σ a smooth surface in N. Pick pâ Σ and r> 0 sufficientlysmall so that we may identify BNr (p) with an Euclidean ball (or half-ball if pâ âN)with the same metric as N. We say that Σi converges locally graphically in the Ck(Ck,α ) topology to Σ at p if for r > 0 sufficiently small we have:(a) Î£â©BNr (p) is the graph of a Ck (Ck,α ) function u defined on Br(0)â TpΣ;(b) Î£iâ©BNr (p) is the graph of Ck (Ck,α ) functions u1i , . . .uLi defined on Br(0)â TpΣfor i sufficiently large and(c) u ji converges to u in the Ck (Ck,α ) topology as iâ â.If L is constant for sufficiently large i, we say that the convergence is L-sheeted.Remark 3.2.4. Let U â N be an open set. If Σi is a sequence of embedded sur-faces converging locally graphically with L-sheets to a two-sided surface Σ ona compacts sets of U then for each compact set â¦ â Î£â©U, r > 0 sufficientlysmall and i sufficiently large we may write Σi â©BNr (â¦) as the graph of functionsui, j : â¦â R, j = 1, . . . ,L, under the exponential map in the normal direction ofΣ. In other words, each connected component Σi, j of Î£iâ©BNr (â¦) can be written as{expx(ui, j(x)NΣ(x)) : x â â¦} for each j = 1, . . . ,L. In addition, ui, j must tend to 0for each j = 1, . . . ,L.443.3 Curvature Estimate for Free Boundary CMCSurfacesIn this section we prove an improvement from uniformly small total curvature es-timate to uniform local pointwise curvature estimate. The proof is inspired by [58,Theorem 1] and we focus on the local estimates around a boundary point. Wepoint out as well that the same proof holds even if the surface is not CMC but hasuniformly bounded C0,α mean curvature.Theorem 3.3.1. Let N be a compact 3-manifold with boundary. There exists asmall enough r0 > 0 such that the following holds: whenever Σ is a properly im-mersed CMC surface in N, Q â Σ, âÎ£â©BNr0(Q) is either empty or free boundaryin âN â©BNr0(Q) and the mean curvature of Σ satisfies HΣ â¤ H0. Then there existsε0 > 0 depending on BNr0(Q) and H0 such that ifâ«Î£â©BNr0 (Q)|A|2 â¤ ε0, thenmax0â¤Ïâ¤r0ï£«ï£­Ï2 supÎ£â©BNr0âÏ (Q)|A|2ï£¶ï£¸â¤C0where the constant C0 only depends on geometry of BNr0(Q) and H0.Proof. Suppose false, that is, for rnâ 0 and Îµnâ 0 there exist free boundary CMCsurfaces Σn â N and Qn â Σn satisfying:(i) Hn â¤ H0;(ii)â«Î£nâ©BNrn (Qn) |An|2 â¤ εn and(iii) max0â¤Ïâ¤rn(Ï2 supÎ£â©BNrnâÏ (Qn) |An|2)> n.Pick 0 < Ïn < rn such thatÏ2n supÎ£â©BNrnâÏn (Qn)|An|2 = max0â¤Ïâ¤rn(Ï2 supÎ£â©BNrnâÏ (Qn)|An|2)and write λ 2n = supÎ£â©BNrnâÏn (Qn) |An|2. For each n there exists zn â Î£â©BNrnâÏn(Qn)such that |An(zn)|> λn2 .45By taking a subsequence we have QnâQ and BNrn(Qn) contained in a geodesicball of N. Without loss of generality we may assume that Î£nâ©BNrn(Qn)â R3 witha metric g. Henceforth we denote by Br(p) the ball in R3 with respect to the metricg.Now, define Î£Ën = Î»n(Î£nâ zn), it satisfies:(a) |AËn(xËn)| â¤ 2 for all xËn â Î£Ënâ©B1(0) and n sufficiently large;(b) |AËn(0)|> 12 and(c)â«Î£Ën |AËn|2 â¤ εn.Indeed, if xËn â B1(0) then xËn = Î»n(xnâ zn) with xn â Î£nâ©B 1λn(zn). It follows thatxn â Î£nâ©Brnâ(Ïnâ 1λn )(Qn). Since(Ïnâ 1Î»n)2supÎ£â©BNrnâ(Ïnâ 1λn )(Qn)|An|2 â¤ max0â¤Ïâ¤rn(Ï2 supÎ£â©BNrnâÏ (Qn)|An|2)= Ï2nÎ»n,it implies that|An(xn)|2 <(11â 1ÏnÎ»n)2Î» 2n .We know that Ï2nÎ» 2n > n thus |An(xn)|< 2λn for n sufficiently large, which proves(a). Property (b) follows from rescaling and (c) holds because the total curvature isscale invariant.Let Î£Ën denote the connected component of Î£Ën â©B1(0) containing 0. We mayfurther assume, after an ambient rotation and translation, that T0Î£Ën = {x3 = 0}.Using property (a), Lemma 3.2.2 implies that there exists rË0 independent of nsuch that Î£Ën â©BrË0(0) is the graph of a function un satisfying âun|B rË04(0)â2,Î± < CË0,where CË0 is independent of n. If 0 is a boundary point then the domain of un is ahalf ball but Lemma 3.2.2 remains true.Finally, un converges, up to a subsequence, in the C2 topology to a function uâ.If we denote its graph by Î£Ëâ then it satisfies |AËâ(0)| â¥ 12 from (b) andâ«Î£Ëâ |AËâ|2 = 0from (c), which is a contradiction and completes the proof of the theorem.463.4 Removable SingularitiesAs we will see later, the compactness result does not give us smooth convergenceeverywhere. The points in which we do not have sufficient curvature estimates arepotential singularities either because of a neckpinching phenomenum where thecurvature may blow up, or self-touching points where the convergence is not singlesheeted. However, we are still able to prove that these are removable singularitiesso the limiting surface is still a smooth object.We are going to prove that if the total curvature is bounded on a CMC surface,then isolated singularities are removable. This is an adaptation of [58, Theorem 2]and the arguments are the same except for the foliation argument to prove unique-ness of the tangent cone. We are going to focus on the case in which the singularityis along the boundary, but the same result holds for interior singularities and theproof follows the exact same arguments.The idea of the proof is to improve the integral curvature bound to a point-wise curvature decay near the singularity to show that the tangent cones are totallygeodesic. By adapting the foliation argument of White [58] we prove that the tan-gent cone is unique from which we can show that near the singularity the surfaceis indeed the graph of a C1 function. We can then improve it further using ellipticregularity.Firstly, let us prove the existence of a local CMC foliation with free boundarywhich will be needed later. This is a straightforward adaptation of [9, Section 3]together with Whiteâs approach to deal with a family of functionals [58, Appendix].Let θ â (0, pi4 ) and define Dθ = {xâR2 : (x1+a)2+x22 â¤ 1 and x1 â¥ 0}, wherea = cosâ1(Î¸) â ( 1â2,1). This is the part of the disk of radius 1 centered on thex1-axis that intersects the line x1 = 0 at angle θ . Its boundary components aredenoted by â0DÎ¸ = âDÎ¸ â©{x âR2 : x1 = 0} and â+DÎ¸ = âDÎ¸ \u00E20Dθ . The regularcylinder over Dθ in R3 is denoted by Cθ = Dθ ÃR, with corresponding boundarycomponents â0CÎ¸ = â0DÎ¸ ÃR and â+CÎ¸ = â+DÎ¸ ÃR.Given a function f âC2,Î±(DÎ¸ ), we define N+g ( f ) to be the normal vector overgraph( f ) with respect to g pointing in the positive direction of the x3-axis, that is,g(N+g ( f ),ââx3 )> 0. We write H+g ( f ) = g(~Hg( f ),N+g ( f )) as the scalar mean curva-ture with respect to N+g ( f ). In particular, ~Hg( f ) points in the positive direction of47x3x2x1âa â+DÎ¸â0DÎ¸CÎ¸Figure 3.1: Blow up picture of a free boundary CMC surface near singularityon the boundarythe x3-axis when H+g ( f )> 0 and in the negative direction otherwise.Let us denote by X the space of C2,α metrics on Cθ and define the mapΦ :RÃRÃXÃC2,Î±(â+DÎ¸ )ÃC2,Î±(DÎ¸ )âC0,Î±(DÎ¸ )ÃC1,Î±(â0DÎ¸ )ÃC2,Î±(â+DÎ¸ )byΦ(h, t,g,w,u) =(H+g (t+u)âh,ââÎ·g(t+u),u|â+DÎ¸ âw),where ηg is the inward conormal vector along â0DÎ¸ .Proposition 3.4.1 ([9, Proposition 21]). For every t0 â R, there exist a neighbour-hood Ut0 of the Euclidean metric δ in X, εt0 > 0 andu : (âÎµt0 ,Îµt0)Ã (t0â εt0 , t0+ Îµt0)ÃUt0ÃBC2,Î± (â+DÎ¸ )εt0 (0)âC2,Î±(DÎ¸ )so that t 7â graph(t + u(h, t,g,w)) defines a C2,α foliation of Dθ Ã [t0â εt02 , t0 +εt02 ] by surfaces with constant mean curvature h with respect to the metric g, freeboundary along â0CÎ¸ and (t+u)|â+DÎ¸ = t+w.48Furthermore, if h > 0 then ~Hg(t +u) points in the positive direction of the x3-axis and in the negative direction when h < 0.Proof. Observe thatΦ defined above is a C1 function and D5Φ(0, t0,δ ,0,0) definesthe same isomorphism as in [9, Appendix B]. The result then follows from theImplicit Function Theorem.Let us denote by B+1 = {x âR3 : âxâ â¤ 1,x1 â¥ 0} the upper half ball in R3 andâ0B+1 = âB+1 â©{x â R3 : x1 = 0}.Theorem 3.4.2. Let g be a Riemannian metric on B+1 and Σ be a smooth, properlyembedded, CMC surface in B+1 {0}, âÎ£â âB+1 , free boundary on â0B+1 {0} and0 â âÎ£. Suppose â«Î£ |AΣ|2 â¤C then Î£âª{0} is a smooth properly embedded CMCsurface in B+1 .Proof. Let r0 > 0 and ε0 > 0 be as in Theorem 3.3.1. Pick δ > 0 sufficiently smallso thatâ«Î£â©B+Î´ |AΣ|2 â¤ ε0. It follows from Theorem 3.3.1 that|AÎ£(x)|dg(x,0)â¤C0,whenever dg(x,0)< δ .Claim 1. Every tangent cone of Σ at 0 is a union of half-planes in R3 {0}.Let riâ â be any sequence and Σi = riΣ its corresponding blow-up around 0.Observe that the curvature estimate above is scale invariant, so Σi satisfies thesame curvature bounds whenever dg(x,0) < riδ . It follows that, up to a subse-quence, Σi converges locally graphically in the C1,α topology on compact sets to acomplete surface Î£â in T0N, which we identify with R3 with the Euclidean met-ric. Lemma 3.2.2 implies that Σi in fact converges locally graphically in the C2,αtopology on compact sets of R3 {0}. In particular, Î£â has free boundary on{x â R3 : x1 = 0}{0} and for any compact set K â R3 {0} we haveâ«Kâ©Î£â|Aâ|2 = limiâââ«Kâ©Î£i|Ai|2 = limiâââ«(râ1i K)â©Î£|AÎ£|2 = 0That is, Aâ = 0. Hence Î£â is an union of half-planes perpendicular to {x â R3 :x1 = 0}.49Claim 2. If δ > 0 is sufficently small then Î£â©B+Î´ {0} is topologically a finite anddisjoint union of disks, half disks or half-disks punctured at 0 with free boundaryon â0B+Î´ {0}.Firstly, we improve the curvature estimates. Fix yâÎ£iâ©B+riÎ´ and put x= râ1i yâÎ£â©B+Î´ . Then |AΣ(x)|dg(x,0) = |Ai(y)|dg(y,0)â 0 as iâ â from the previousclaim. Thus limxâ0 |AΣ(x)|dg(x,0) = 0.Secondly, we use a standard Morse Theory argument. Let f : Î£â©B+Î´ â R bedefined by f (x) = 12 dg(x,0)2. We can write its Hessian at a point x and direction vas Hess f (v,v)x =AΣ(v,v)xg(NΣ(x), Î³Ë(1))x+Qx(v,v), where NΣ is the normal vectorfield on Σ, γ is the minimizing geodesic in B+1 from 0 to x and Q is a quadratic formsatisfying Qâ¥ 12 g as long as dg(x,0)< δ is sufficiently small. Hence, any criticalpoint of f in a small neighbourhood of 0 is a strict local minimum, even if thecritical point is along the boundary. It follows from Morse Theory for manifoldswith boundary [40] that every connected component of Î£â©B+Î´ is a disk with asingle critical point, a free boundary half disk with a single critical point or a freeboundary half disk punctured at 0 without critical points. Since critical pointscannot accumulate, it must be finite for δ sufficiently small. This proves the claim.Next, we shall prove that for each punctured half-disk its tangent cone is unique.Pick Î£Ëâ Î£â©B+Î´ {0} a connected component that corresponds to a punctured half-disk. Let Î£Ëi be the corresponding dilation by riâ â and Î£Ëâ â Î£â its tangent coneat 0. As we have seen above, Î£Ëâ is a half-plane perpendicular to {x âR3 : x1 = 0}.Claim 3. The tangent cone Î£Ëâ is unique, that is, it is independent of the blow-upsequence riâ â.Without loss of generality let us identify Î£Ëâ with the half-plane P+ = {x âR3 : x3 = 0}â© {x â R3 : x1 â¥ 0}. We may also assume, possibly taking anothersubsequence, that the mean curvature vector of Î£Ëi points in the positive directionof the x3-axis for i sufficiently large.Let Dθ and Cθ be as in Proposition 3.4.1 Since Î£Ëi is embedded, then for isufficiently large Î£Ëi â© â+CÎ¸ is the graph of a function wi = wri over â+DÎ¸ . Weknow that Î£Ëi converges in C2,α graphical sense to P+ away from zero, from whichfollows that âwiâ2,Î± â 0 where the norm is defined on â+DÎ¸ .50The mean curvature of Î£Ëi is given by HËi = riâ1HÎ£ which is constant at eachi and tends to 0 as i tends to infinity. If we denote by gi the corresponding blowup of the metric g in a neighbourhood of 0 in B+1 , it follows from Proposition3.4.1 that for all i sufficiently large there exists ε0 > 0 and a unique function ui,t =u(HËi, t,gi,wi) : Dθ âR for each |t|< ε0 such that ui,t = t+wi on â+DÎ¸ , the graphof ui,t over Dθ meets â0CÎ¸ orthogonally, it has constant mean curvature equal to HËiand its mean curvature vector points in the same direction as the mean curvaturevector of Î£Ëi. Furthermore, ui,t varies smoothly on t and it defines a foliation of aregion Dθ Ã [âc,c] for some c > 0 independent of i.Let ti â (âÎµ0,Îµ0) be such that ui,ti(0) = 0. Since their mean curvature vectorpoints in the same direction, we may apply the maximum principle to the graph ofui,ti and Î£Ëi to conclude that Î£Ëi must be contained entirely to one side of the graphof ui,ti . The side itself will depend on whether ti is positive or negative. In case themean curvature vector of Î£Ëi were to point in the negative direction of the x3-axis,we would have used the same argument with u(âHËi, t,gi,wi) instead.Finally, pick another sequence râ²iââ and let Î£Ëâ²i, Pâ²+ be its corresponding blow-up and tangent cone respectively. For each k we pick ik so that râ²ik > krk. Observethat Î£Ëik =râ²ikrkÎ£Ëk which is contained to one side of the graph ofrâ²ikrkuk,tk . Since eachuk,tk is regular at 0,râ²ikrkuk,tk must converge to a unique tangent cone, that is, P+. IfPâ²+ were different from P+, then it would imply thatrâ²ikrkÎ£Ëk contains points on bothsides of the graph ofrâ²ikrkuk,tk . Notice that Pâ²+ must also be a half plane with freeboundary on {x â R3 : x1 = 0}. This concludes the claim.It follows from uniqueness of the tangent cone that Î£Ëâª{0} is the graph of aC1 function u around a neighbourhood of 0 over the tangent cone. Since u is also asolution to the CMC equation with smooth coefficients, then by elliptic regularityit must also be smooth.To conclude the proof, now that we know that Î£Ëâª {0} is regular, it followsfrom the maximum principle that it is the unique connected component containing0. Thus Î£âª{0} is properly embedded.We now state the equivalent result for removable interior singularities withoutrepeating the proof.51Theorem 3.4.3. Let g be a Riemannian metric on B1 and Σ be a smooth, properlyembedded, CMC surface in B1{0}with HÎ£â¤H0, and 0âÎ£. Supposeâ«Î£ |AÎ£|2â¤Cthen Î£âª{0} is a smooth properly embedded CMC surface in B1.Remark 3.4.4. The proof is exactly the same with a minor modification in theconstruction of the foliation. That is, by dropping the Neumann component in thedefinition of Φ in Proposition 3.4.1.3.5 Compactness TheoremIn this section we will prove our main theorem for free boundary embedded CMCsurfaces. As we shall see, the limiting surface may not be embedded because CMCsurfaces may have tangential self-intersection as long as the normal vector pointsat opposite directions.Theorem 3.5.1. Let N be a compact 3-dimensional manifold with boundary. Sup-pose HâN â¥H0 and let Σi be a sequence of free boundary embedded CMC surfaceswith mean curvature Hi, genus gi and number of ends ri satisfying:(a) |Hi| â¤ H0;(b) gi â¤ g0;(c) ri â¤ r0;(d) area(Î£i)â¤ A0 and(e) length(âÎ£i)â¤ L0.Then there exists a smooth properly almost embedded CMC surface Σ â Nand a finite set Î â Σ such that, up to a subsequence, Σi converges to Σ locallygraphically in the Ck topology on compact sets of N \u00CE for all k â¥ 2. Moreover, ifΣ is minimal then it is properly embedded.If in addition (N,âN) satisfies either RicN > 0 and AâN â¥ 0 or RicN â¥ 0 andAâN > 0, then:(i) when HΣ = 0 the convergence is at most 2-sheeted;52(ii) when HΣ 6= 0, then the convergence is 1-sheeted away from Î.Proof. Let us dente by Ai the second fundamental form of Σi. Given x â Σ, itfollows from the Gauss equation that |Ai|2(x) = H2i + 2KN(TxÎ£)â 2Ki(x), whereKN ,Ki are the sectional curvatures of N and Σi respectively. From the Gauss-Bonnet theorem we haveâ«Î£i|Ai|2 = H2i area(Î£i)+2â«Î£iKN(TxN)+2â«âÎ£iÎºg+4pi(2gi+ riâ2),where κg denotes the geodesic curvature of âÎ£i. Because Σi is free boundary, wehave that that κg = AâN(ÏâÎ£,ÏâÎ£), where AâN is the second fundamental form ofâN with respect to the inner normal vector and ÏâÎ£ is the unit tangent vector of âÎ£.Since N is compact, there exists a constant C =C(N)> 0 such thatâ«Î£i|Ai|2 â¤C(H2i area(Σi)+gi+ ri+ area(Σi)+ length(âÎ£i)).Hence, from hypotheses (a)-(e)we have that the total curvature is uniformly boundedby a constant C0 =C0(N,H0,g0,r0,A0,L0)> 0.Denote by µi the Radon measure on N defined by µi(U) =â«Î£iâ©U |Ai|2, for asubset U â N. It follows from the above that there exists a Radon measure µ inN such that, up to a subsequence, µi converges weakly to µ . Furthermore, the setÎ= {p â N : µ({p})â¥ 1} has at most C0 elements.For each x â N \u00CE there exists r > 0 such that µ(BNr (x))< 1. Hence, for eachi sufficiently large µi(BNr (x))< 1, that is,â«Î£iâ©BNr (x) |Ai|2 < 1. By possibly choosinga smaller value of r, it follows from Theorem 3.3.1 thatsupÎ£iâ©BNr2|Ai|2 â¤C,for some constant C > 0 independent of i.Let r < r0 as in Lemma 3.2.2 and suppose that, up to a subsequence, Î£iâ©BNr4(x)is non-empty for all i sufficiently large. Since Σi is embedded then Σi â©BNr4(x) isthe union of disjoint embedded connected components Σi,1, . . . ,Σi,L each of whichis the graph of a function defined on an open ball of fixed radius on Tyi, jΣi, j forsome yi, j â Σi, j, j = 1, . . . ,L. Because Σi is compact, the number of sheets L must53be finite and thus constant for i sufficiently large. Hence, under the appropriateidentifications we may further assume that Σi, j â©BNrâ² (x) is the graph of a functionui, j defined on Brâ²(0)â TyiΣi, for some 0 < râ² < r8 depending only on L and a fixedyi â Î£iâ©BNrâ² (x). Furthermore ui, j has uniform C2,α bounds as in Lemma 3.2.2.We may assume that, up to a subsequence, yi converges to yâ² and TyiΣi con-verges to a plane P â Tyâ²N. In which case, under further identifications, we havethat Σi, j â©BNrâ²2(yâ²) is the graph of a function uâ²i, j defined on an open ball on P (orhalf-ball in case yâ² is on the boundary of N) and uniform C2,α estimates for all isufficiently large, and each j = 1, . . . ,L. Hence, up to a subsequence uâ²i, j convergesto a function uâ²j in the C2,β topology for all β < α and Σi â©BNrâ²2(yâ²) converges toÎ£â²j = graph(uâ²j) for each j = 1, . . . ,L. From this it follows that yâ² â Î£â² = âª jÎ£â²j andP is in fact Tyâ²Î£â².Now, given any compact set K â N \u00CE we cover it by finitely many open balls{BNr4(xk)}k=0,...,m as above so that, up to a subsequence, Î£iâ© (âªkBNr4(xk)) convergesto a surface Î£â² â N \u00CE locally graphically in the C2,β topology on K. Taking acountable exhaustion by compact sets and using a diagonal argument we have that,up to a subsequence, Σi converges to Î£â² locally graphically in the C2,β topology oncompact sets of N \u00CE. Smooth convergence away from Î follows from Allardâsregularity Theorem [5, 6, 36].Since Σi is a properly embedded CMC surface with free boundary along âN\u00CE,then Î£â² is a properly almost embedded CMC surface in N \u00CE with free boundaryalong âN \u00CE. Observe that on a neighbourhood of any self-touching point of Î£â² thesurface can be written as connected components that lie to one side of one another.A transversal self-intersection is an open condition so it would contradict the factthat Σi is embedded.Define Σ to be the closure of Î£â², so Î=Î£\u00CE£â². Since Î is finite, there exists r> 0so that BNr (p) {p} contains no points of Î. From the graphical convergence oncompact sets of BNr (p){p}, each sheet must converge to an embedded componentof Î£â©BNr (p) {p}. Since there are only finitely many sheets, we may pick r > 0sufficiently small so that every component of Î£â© BNr (p) {p} contains p in itsclosure.Claim 1. The limit surface Σ is regular.54We know thatâ«Î£â©(BNr (p){p}) |AΣ|2 â¤ C0. Thus we may apply the RemovableSingularity Theorems 3.4.2 or 3.4.3 to each embedded component of Î£â© (BNr (p){p}) depending on whether p belongs to the boundary of N or to the interior. HenceΣ= Î£â²âªÎ is a regular surface.Now suppose that HΣ = 0. Then the maximum principle for minimal surfacesimplies that there are no self-touching points, so Σ is embedded.We now prove properties (i) and (ii). Henceforth, let us assume that eitherRicN > 0 and AâN â¥ 0 or RicN â¥ 0 and AâN > 0.The following argument is the same as in [51] and [9] and we include the mainidea without repeating the calculations.Claim 2. If HΣ 6= 0 then the convergence is 1-sheeted away from Î.Suppose by contradiction that Σi converges to Σwith at least 2 sheets. Since theconvergence is graphical over compact sets of N \u00CE, for any compact set â¦â Σ\u00CEand a sufficiently small tubular neighbourhood Vr(â¦) of â¦, Î£iâ©Vr(â¦) contains atleast 2 connected components Σ0i and Σ1i each of which can be written as follows,for i sufficiently large:Σνi = {expp(uνi (p)NΣ(p)) : p ââ¦},for uνi : â¦â R, ν = 0,1.We may assume without loss of generality that u1i > u0i for all i sufficientlylarge since the surface Σi is embedded.Let us denote vi(t) = u0i + t(u1i â u0i ) and Φi(p, t) = expp(vi(t)(p)NΣ(p)) forp â â¦. Consider the variation Σi(t) = {Φi(p, t) : p â â¦} so that Σi(ν) = Σνi , ν =0,1. Take a function w â Câc (â¦) with compact support and define wν â Câc (Σνi )by wν(Φ(p,ν)) = u(p) for each ν = 0,1. Now, consider any compactly supportedvector field X in Vr(â¦) such that g(X ,NΣνi ) =wν along Σνi , for each ν = 0,1, and itsassociated flow Ψ(q,s) on Vr(â¦). Finally we consider the following 2-parametervariation:Σi(t,s) = (Ψs)]Σi(t).55From the first variation formula it follows thatââ sâ£â£â£â£s=0area(Î£i(Î½ ,s)) =âHiâ«Î£Î½iwÎ½ ,for each ν = 0,1. Since the mean curvature vectors of each sheet converge to ~HΣ,we have that their normal vectors point in the same direction for i sufficiently large.That is, their scalar mean curvature has the same sign, so we haveââ sâ£â£â£â£s=0area(Î£i(1,s))â ââ sâ£â£â£â£s=0area(Î£i(0,s)) =âHiâ«Î£(JÎ¦1i â JΦ0i )w,where for each ν = 0,1, Φνi (p) =Φi(p,ν) and JΦνi its corresponding Jacobian.Following the same idea and calculations as in [9], using the Mean Value The-orem and taking the second variation of area, for some ti â (0,1), we obtain thathËi = u1i âu0i satisfiesââ«â¦w(LihËi+HiFi(hËi))+â«âNâ©â¦Bi(w, hËi) = 0.Here Li is an elliptic operator in divergence form, Li converges uniformly to theJacobian operator LΣ of Σ as iââ, Fi(hËi) = (JΦ1i âJÎ¦0i ) and Bi(w1,w2) convergesuniformly to w1( âw2âÎ· +AâN(NÎ£,NÎ£)w2), where η denotes the outward conormalvector of âÎ£ and AâN the second fundamental form with respect to the outwardconormal of âN. Since uνi tends to 0 in C2, Φνi tends to the identity uniformly; thatis, Fi(hËi) tends to 0.Fix a point q0 â â¦ and define hi(p) = hËi(p)hi(q0) . Using Harnack estimates and ablow-up contradiction argument we have that hi converges smoothly to a functionh on â¦ satisfying ï£±ï£´ï£²ï£´ï£³LÎ£h = 0, on â¦;âhâÎ·+AâN(NÎ£,NÎ£)h = 0, on âÎ£â©â¦.Since the blow up of the elliptic equation for hi is the one above, the proof is thesame as in [9, Claim 1, p.20]. It follows from the maximum principle for ellipticequations that h > 0 on â¦. By taking an exhaustion of Σ Î by compact sets56and using a diagonal sequence argument, we have that hi converges to h > 0 oncompact sets of Σ\u00CE. Again, it follows from the exact same argument as in [9] thath is uniformly bounded on Σ\u00CE, thus it extends to a smooth function on Σ, whichis positive by the maximum principle.Finally, we observe that the conditions RicN > 0 and AâN â¥ 0 or RicN â¥ 0 andAâN > 0 imply that the quadratic formQΣ(w1,w2) =ââ«Î£w1LÎ£w2+â«âÎ£w1(âw2âÎ·+AâN(NÎ£,NÎ£)w2)is positive definite. Hence its first eigenvalue is positive. However, h is a positiveeigenfunction so it must correspond to the first eigenvalue, which is a contradictionsince the corresponding eigenvalue is 0. We conclude that the convergence mustbe 1-sheeted away from Î.The case HΣ = 0 follows the exact same arguments as above. Unlike the pre-vious case, the mean curvature vector of each graph converges to 0 so they mayhave opposite orientation. As a consequence, the scalar mean curvature may haveopposite sign, hence the function Fi would not tend to 0 as desired.Claim 3. If HΣ = 0 then the convergence is at most 2-sheeted.Suppose the convergence is at least 3-sheeted. Then at least two sheets havethe same orientation; that is, the normal vectors of the sheets point in the samedirection. Following the same argument as in the previous case, the functions Fiwill converge to 0, so we can construct a positive eigenfunction h> 0 for the Jacobioperator LΣ on Σ corresponding to the eigenvalue 0. This is a contradiction sinceLΣ is a positive operator under either of the conditions RicN > 0 and AâN â¥ 0 orRicN â¥ 0 and AâN > 0.This completes the proof.57Chapter 4Higher Dimensional Surgery andSteklov Eigenvalues4.1 IntroductionIn this chapter we study questions related to shape optimization for the Stekloveigenvalue problem on compact Riemannian manifolds of dimension at least three.The classical theorem of R. Weinstock [57] states that among all simply connecteddomains in R2 with fixed boundary length 2pi , the unit disc uniquely maximizesthe first Steklov eigenvalue. It was shown in [13] that the Weinstock inequalityholds in any dimension, provided one restricts to the class of convex domains.Namely, for every bounded convex domain â¦ â Rn we have ÏÂ¯1(â¦) â¤ ÏÂ¯1(Bn),where ÏÂ¯ j(â¦) = Ï j(â¦)|ââ¦|1/(nâ1) denotes the normalized eigenvalues. However,in the wider class consisting of contractible domains, the higher dimensional ana-logue of Weinstockâs theorem fails since there exist contractible domains Bnε,δ (seeFigure 4.1) with ÏÂ¯1(BnÎµ,Î´ ) > ÏÂ¯1(Bn) when n â¥ 3 ([31]). The proof involves a 1-dimensional surgery in which a small tubular neighbourhood of a curve connectingboundary components is removed from an annular domain. The construction in[31] shows more generally that in dimensions greater than or equal to three thenumber of boundary components does not affect the supremum of the normalizedfirst Steklov eigenvalue. This is in contrast to the situation in dimension two, whereby adding an extra boundary component to a surface the normalized first Steklov58eigenvalue can be made strictly larger ([30], [42]).In this chapter we consider the effect of higher dimensional surgeries on theSteklov spectrum of compact Riemannian manifolds with boundary. Specifically,we show that one can perform surgeries of codimension two or higher while keep-ing the normalized Steklov eigenvalues nearly unchanged. Given a compact n-dimensional Riemannian manifold â¦ with boundary, a compact properly embed-ded m-dimensional submanifold Σ of â¦, and δ > 0 small, we let â¦Î£,Î´ denote theLipschitz domain obtained by removing the δ tubular neighbourhood of Σ from â¦.We call this procedure a surgery of codimension nâm (see Section 4.2 for moreprecise details). Our main result is the following.Theorem 4.1.1. The Steklov spectrum of a compact Riemannian manifold withboundary changes continuously under surgeries of codimension at least two, in thesense that, if nâmâ¥ 2 then limÎ´â0+ ÏÂ¯ j(â¦Î£,Î´ ) = ÏÂ¯ j(â¦) for j = 0,1,2, . . ..One purpose of such surgeries is to simplify the topology of the manifold.When m = 1 and j = 1 this theorem was proved in [31], in which case the surgerycan be applied to construct a manifold with connected boundary from any givenmanifold with boundary, while keeping the first normalized Steklov eigenvaluenearly unchanged. Theorem 1.1 implies more generally that the supremum of the j-th normalized eigenvalue among all manifolds is the same as the supremum amongmanifolds having relatively simple topology; that is, among manifolds which canbe obtained by performing surgeries up to codimension two. Specifically, our resultimplies that given any compact Riemannian manifold â¦ of dimension n â¥ 3, andgiven any ε > 0 and k â N, there exists a smooth subdomain â¦Ë of â¦ such that thehomomorphism of the m-th homology groups iâ : Hm(â â¦Ë)â Hm(â¦Ë) induced bythe inclusion i : â â¦Ë âªâ â¦Ë is injective for m = 0,1, . . .min{nâ 3,ân2â}, and suchthat |ÏÂ¯ j(â¦Ë)â ÏÂ¯ j(â¦)|< ε for j = 1, . . . ,k. This conclusion involves some standardbut rather involved algebraic topology, and we omit the proof since our results hereare analytic in nature.An immediate consequence of Theorem 4.1.1 is that for n â¥ 3, balls do notmaximize higher normalized Steklov eigenvalues among contractible domains inRn.59Corollary 4.1.2. Let n â¥ 3. For any k â¥ 1, there exists a contractible domain â¦âin Rn such thatÏÂ¯ j(â¦â)> ÏÂ¯ j(Bn)for j = 1, · · · ,k.This is not surprising, especially in light of the fact that a disc does not max-imize higher Steklov eigenvalues among simply connected domains in R2. Theclassical result [38] gives the upper bound ÏÂ¯ j(â¦)â¤ 2pi j for any simply connecteddomain â¦ in R2 and all j â¥ 0. When j = 1, equality is characterized in Wein-stockâs theorem ([57]), but for j â¥ 2 the inequality is strict ([34], [32]). However,it was shown in [34] that the inequality is sharp and is achieved in the limit by asequence of simply connected domains degenerating to j identical discs. By anal-ogy with the n= 2 case, it is natural to ask whether a similar result is true in higherdimensions. Using ideas from Theorem 4.1.1 we show that this is not the case.Theorem 4.1.3. For nâ¥ 3 and jâ¥ 2, the supremum of the j-th normalized Stekloveigenvalue among contractible domains in Rn is not achieved in the limit by asequence of contractible domains degenerating to the disjoint union of j identicalround balls.This is a direct consequence of a more general theorem on the convergenceof Steklov eigenvalues of overlapping domains as they are pulled apart, Theorem4.3.2, discussed in Section 4.3.The chapter is organized as follows. In section 4.2 we recall basic knowl-edge about the Steklov eigenvalue problem and prove our main result, Theorem4.1.1, on higher dimensional surgeries and the Steklov spectrum. In section 4.3 wediscuss applications and related results in connection with questions about shapeoptimization for the Steklov problem for domains inRn, and prove Corollary 4.1.2,Theorem 4.3.2, and Theorem 4.1.3.4.2 Continuity of Steklov Eigenvalues underCodimension 2 SurgeriesIn this section, we recall some basic facts about the Steklov eigenvalue problem,and prove our main theorem on the continuity of eigenvalues under certain higher60dimensional surgeries of at least codimension 2. Let (â¦,g) be a compact, con-nected n-dimensional smooth Riemannian manifold with nonempty boundary. TheSteklov eigenvalue problem is given byï£±ï£²ï£³âu = 0 in â¦âuâÎ½ = Ïu on ââ¦where ν is the outer unit normal to â¦. When the trace operator T : H1(â¦)âL2(ââ¦) is compact, in particular, if the domain has Lipschitz boundary, the spec-trum of the Steklov eigenvalue problem is discrete0 = Ï0 < Ï1(â¦)â¤ Ï2(â¦)â¤ Â·Â· Â·and the eigenvalues have a standard Rayleigh quotient characterizationÏ j(â¦) = inf{â«â¦ |âu|2â«ââ¦ u2: 0 6= u â H1(â¦),â«ââ¦uÏi = 0 for i = 0, . . . , jâ1}(4.2.1)where {Ï0,Ï1,Ï2, . . .} is a complete orthonormal basis of L2(ââ¦) such that Ïi is aneigenfunction with eigenvalue Ïi(â¦) for each i = 1, 2, . . .. Alternatively,Ï j(â¦) = infE j+1supuâE j+1{0}â«â¦ |âu|2â«ââ¦ u2(4.2.2)where the infimum is taken over all ( j + 1)-dimensional subspaces E j+1 of theSobolev space H1(â¦).Definition 4.2.1. Given a smooth m-dimensional submanifold Σ embedded in themanifold â¦, with boundary âÎ£ embedded in ââ¦, and meeting ââ¦ orthogonallyalong âÎ£. LetLδ = {x ââ¦ : d(x,Σ)< δ} and Tδ = {x ââ¦ : d(x,Σ) = Î´}.Denoteâ¦Î´ =â¦LÎ´ .61Taking out the tubular neighborhood Lδ of Σ gives a domain â¦Î´ . Throughout thischapter, we call this procedure surgery of dimension m or surgery of codimensionn-m.We will apply the following âno concentration" lemma to show that if 1â¤mâ¤nâ 2, then the interior and boundary L2 norms of a sequence of eigenfunctionsdonât concentrate near the neck Tδ as δ â 0+. When m = 1, that is when Σ is asmooth curve, this phenomenon was verified in [31, Lemma 4.2]. We prove herethat it is in fact still true provided 2â¤ mâ¤ nâ2.Lemma 4.2.2. Suppose 1â¤ mâ¤ nâ2. There exists a constant r0 > 0 such that ifuδ âW 1,2(â¦Î´ ) and â«â¦r0/2\u00E2¦r0u2Î´ +â«â¦Î´ \u00E2¦r0|âuÎ´ |2 â¤Cfor δ â (0,r0/2), with C independent of δ . ThenlimÎ´â0âuÎ´âL2(TÎ´ ) = 0, (4.2.3)limsâ0âuÎ´âL2(â¦Î´ \u00E2¦s) = 0, (4.2.4)limsâ0âuÎ´âL2((LsLÎ´ )â©ââ¦) = 0 (4.2.5)for any 0 < δ < s < r0/2.Proof. For simplicity of presentation we will assume that the normal bundle of Σ istrivial; otherwise, we can consider local trivializations and add up all of the corre-sponding estimates. Choose r0 > 0 sufficiently small such that the exponential mapof the normal bundle of Σ is a diffeomorphism from the r0 neighbourhood of thezero section onto its image, and such that the metric on the r0 tubular neighbour-hood Lr0 of Σ in M is uniformly equivalent to the product metric gË+dr2+r2gSnâmâ1on Î£ÃDr0 , where gË is the metric on Σ induced from g, gSnâmâ1 is the standard met-ric on the sphere Snâmâ1 and Dt is the ball of radius t centred at the origin in Rnâm.Since Σ intersects ââ¦ orthogonally along âÎ£, we may further choose r0 sufficiently62small such that the metric on Lr0â©ââ¦ is uniformly equivalent to the product metricgâÎ£+dr2+ r2gSnâmâ1 .We first localize the support of uδ to lie near Σ. Choose a smooth radial cut-offfunction such thatÏ(x,r,Î¸) =ï£±ï£²ï£³1 r â¤r020 r â¥ r0and define vδ = ÏuÎ´ . It follows from the Cauchy-Schwarz and arithmetic geomet-ric mean inequalities that|âvÎ´ |2 â¤ 2(Ï2|âuÎ´ |2+u2δ |âÏ|2).Since |âÏ| can be bounded in terms of r0 which is a fixed number, we haveâ«â¦Î´|âvÎ´ |2 â¤ 2â«â¦Î´ \u00E2¦r0|âuÎ´ |2+2C1â«â¦r0/2\u00E2¦r0u2Î´ â¤C2 (4.2.6)where C1, C2 are constants depending only on C and r0. We will choose δ muchsmaller than r0. From the construction we have that uδ = vδ on Tδ , and thus toprove equation (4.2.3) in the lemma it suffices to prove that for any ε > 0â«TÎ´u2Î´ =â«TÎ´v2Î´ â¤ Îµâ«â¦Î´|âvÎ´ |2for sufficiently small δ . Since the metric on â¦Î´ is uniformly equivalent to theproduct metric on the support of vδ , it suffices to prove the estimate for the productmetric.For a fixed point p in Σ we denote the restriction of vδ (x,r,θ) to the annulusDr0 Dδ in Rnâm at this point p by v(r,θ). Choose a harmonic function h onDr0 Dδ as follows ï£±ï£´ï£´ï£´ï£²ï£´ï£´ï£´ï£³âh = 0 Dr0 Dδh = v = 0 âDr0h = v âDÎ´ .63Since harmonic functions minimize the Dirichlet energy we haveâ«Dr0DÎ´|âh|2 â¤â«Dr0DÎ´|âv|2. (4.2.7)For any Ï with δ â¤ Ï â¤ r0 we haveâ«Dr0DÏâh2 =â«âDr0âh2â rââ«âDÏâh2âÏ=ââ«âDÏâh2âÏ=âÏnâmâ1 ddÏ[Ïân+m+1â«âDÏh2]where last equality follows since the volume measure on âDÏ is Ïnâmâ1 times thaton the unit sphere âD1. Since âh = 0, we haveâ«Dr0DÏâh2 = 2â«Dr0DÏ|âh|2,which together with (4.2.7) impliesâÏnâmâ1 ddÏ[Ïân+m+1â«âDÏh2]= 2â«Dr0DÏ|âh|2â¤ 2â«Dr0DÎ´|âh|2â¤ 2â«Dr0DÎ´|âv|2.Now dividing both sides by Ïnâmâ1 and integrating with respect to Ï over theinterval [δ ,r0] we obtainÎ´ân+m+1â«âDÎ´v2 â¤ 2(â« r0Î´Ïân+m+1 dÏ)â«Dr0DÎ´|âv|2.If 1â¤ mâ¤ nâ2, this implies thatâ«âDÎ´v2 â¤ εn(δ )â«Dr0DÎ´|âv|264where εm+2(δ ) = 2δ ln( r0δ ) and εn(δ ) =2Î´nâmâ2 for nâ¥m+3. Integrating the aboveinequality over Σ we obtainâ«TÎ´v2Î´ â¤ εn(δ )â«Î£mâ«Dr0DÎ´|âv|2 â¤ εn(δ )â«â¦Î´|âvÎ´ |2, (4.2.8)since |âv| â¤ |âvÎ´ | because âv is the Euclidean gradient on the slice Dr0 Dδ at apoint on Σ. Since limÎ´â0 εn(δ ) = 0 for 1â¤ mâ¤ nâ2, this completes the proof of(4.2.3).In what follows we shall prove (4.2.4) and (4.2.5) of the lemma. By the sameargument used to prove (4.2.8), we have that for any δ â¤ t < r0/2â«Ttv2Î´ â¤ Îµn(t)â«â¦Î´|âvÎ´ |2 â¤C2Îµn(t)where εn(t) is defined as above, and C2 is as in (4.2.6) and is independent of δ .Integrating with respect to t over [δ ,s] for s < r0/2 yieldsâ«â¦Î´ \u00E2¦sv2Î´ =â«LsLÎ´v2Î´ â¤ï£±ï£²ï£³C2(s2 ln( r0s )âÎ´ 2 ln( r0δ )+ s2âÎ´ 22), n = m+2C2 (s2âÎ´ 2)/(nâmâ2), nâ¥ m+3.(4.2.9)Since the right hand sides tend to zero as sâ 0 for all δ < s, this concludes theproof of (4.2.4).Denote Σt = {xâ Σ : distÎ£(x,âÎ£)â¤ t} and denote âÎ£t = {xâ Σ : distÎ£(x,âÎ£) =t} (note this is only part of the topological boundary of Σt). We choose t sufficientlysmall such that HâÎ£Ï â¤ C for all Ï â¤ t, where HâÎ£Ï is the mean curvature of âÎ£Ïwith respect to outer unit normal. By the coarea formula,â« t0â«âÎ£Ï(â«DsDÎ´ (x)v2Î´)dÏ =â«Î£t(â«DsDÎ´ (x)v2Î´)â¤â«Î£â«DsDÎ´ (x)v2δ =â«LsLÎ´v2Î´ .DenoteFs,δ (Ï) :=â«âÎ£Ï(â«DsDÎ´ (x)v2δ).65Then (4.2.9) yieldslimsâ0â« t0Fs,δ (Ï) dÏ = 0.Thus there exists Ï0 â (0, t) such thatlimsâ0Fs,Î´ (Ï0) = 0.We differentiate Fs,δ (Ï) to getddÏFs,Î´ (Ï) =â«âÎ£Ï(â«DsDÎ´ (x)2vÎ´dvÎ´dÏ)ââ«âÎ£Ï(â«DsDÎ´ (x)v2Î´)HâÎ£Ï .Integrating over [0,Ï0], applying the coarea formula, and using Cauchy-Schwarzinequality and the arithmetic-geometric mean inequality result in that for any 0 <ε < 1|Fs,δ (Ï0)âFs,Î´ (0)|=â£â£â£â£â£â«Î£Ï0(â«DsDÎ´ (x)2vÎ´dvÎ´dÏ)ââ« Ï00â«âÎ£Ï(â«DsDÎ´ (x)v2Î´)HâÎ£Ï dÏâ£â£â£â£â£â¤ Îµâ«Î£Ï0(â«DsDÎ´ (x)|âvÎ´ |2)+(1/Îµ+C)â«Î£Ï0(â«DsDÎ´ (x)v2Î´)â¤C2Îµ+(1/Îµ+C)â«LsLÎ´v2Î´ .Taking ε = s and letting s tend to zero, from (4.2.9) it follows thatFs,δ (0)â 0 as sâ 0.Hence we obtain the equation (4.2.5).Lemma 4.2.3. Consider the following logarithmic cut-off function along the sub-manifold Σm, where 1â¤ mâ¤ nâ2,ÏÎ´ =ï£±ï£´ï£´ï£´ï£²ï£´ï£´ï£´ï£³0 r â¤ δ 22lnÎ´âlnrlnÎ´ δ2 â¤ r â¤ δ1 δ â¤ r.66Thenâ«â¦ |âÏÎ´ |2â 0 as δ â 0+.Proof. Integrating with respect to product metric givesâ«â¦|âÏÎ´ |2 â¤â«Î£â«DÎ´ DÎ´2|âÏÎ´ |2 =c(n)|Σ|(lnδ )2â« δδ 2rnâmâ3 dr = c(n)|Σ|δ (n)where δ (n) =â1/ lnδ when n=m+2 and δ (n) = (δ nâmâ2âÎ´ 2nâ2mâ4)/(nâmâ2)(lnÎ´ )2 when n >m+2. Since δ (n)â 0 as δ â 0, this completes the proof.We now prove our main theorem, that given a compact Riemannian mani-fold with boundary (â¦n,g), surgeries of dimension m (see Definition 4.2.1) canbe performed while keeping the Steklov eigenvalues nearly unchanged, provided1â¤ mâ¤ nâ2.Theorem 4.1.1. If 1â¤mâ¤ nâ2, then limÎ´â0+ Ï j(â¦Î´ ) = Ï j(â¦) for j = 0,1,2, . . ..Proof. Let u0δ ,u1δ ,u2δ , . . . be L2(ââ¦Î´ )-orthonormal Steklov eigenfunctions such thatu jδ is a Steklov eigenfunction of â¦Î´ with eigenvalue Ï j(â¦Î´ ),ï£±ï£²ï£³âujÎ´ = 0 in â¦Î´âu jÎ´âÎ· = Ï j(â¦Î´ )ujδ on ââ¦Î´ .Claim 1. For any j â N, Ï j(â¦Î´ ) is uniformly bounded from above for small δ .Proof. Let { f1, . . . , f j+1} â H1(â¦) be j+ 1 functions on â¦ with support in â¦r0(where r0 is as in Lemma 4.2.2), that are linearly independent on ââ¦. Let E =span{ f1, · · · , f j+1}. Then if δ < r0, any function in E is a valid test function forthe min-max variational characterization (4.2.2) of Ï j(â¦Î´ ), and soÏ j(â¦Î´ )â¤ supuâE{0}â«â¦Î´ |âu|2â«ââ¦Î´ u2 = supuâE{0}â«â¦r0|âu|2â«ââ¦r0u2â¤ Î jwhere Î j is independent of δ .Since u jδ is a Steklov eigenfunction of â¦Î´ with eigenvalue Ï j(â¦Î´ ),â«â¦Î´|âu jδ |2 = Ï j(â¦Î´ )â«â¦Î´(u jδ )2 = Ï j(â¦Î´ )â¤ Î j. (4.2.10)67By (4.2.10) and since âu jÎ´âL2(ââ¦Î´ ) = 1, by standard theory,âu jÎ´âL2(K) â¤C(Î j,K) (4.2.11)for any compact subset K of â¦\u00CE£.Elliptic boundary estimates ([33, Theorem 6.29]) give boundsâu jÎ´âC2,Î± (K) â¤Câu jÎ´âC0(K)for any compact subset K ofâ¦\u00CE£ for all sufficiently small δ , where C=C( j,Î±,Î j,K).By Sobolev embedding and interpolation inequalities ([1, Theorem 5.2], [33, (7.10)]),âu jÎ´âC0(K) â¤C(Îµâu jÎ´âC2(K)+ Îµâµâu jÎ´âL2(K))where ε > 0 can be taken arbitrarily small, µ > 0 depends on n, and C depends onK. Hence âu jÎ´âC2,Î± (K) â¤C with C independent of δ . By the Arzela-Ascoli theoremand a diagonal sequence argument, there exists a sequence Î´iâ 0 such that for allj, u jδi converges in C2(K) on compact subsets K ââ¦\u00CE£ to a harmonic function u jon â¦\u00CE£, satisfyingâu jâÎ·= Ï ju j on ââ¦\u00E2Î£with Ï j = limiââÏ j(â¦Î´i).Claim 2. For each j â¥ 1 â N, u j can be extended to a Steklov eigenfunction of â¦with eigenvalue Ï j.Proof. First observe that u j â H1(â¦ \u00CE£). Fix the compact subset â¦Î´N for somelarge N. Then âu jÎ´iâ2L2(â¦Î´i ) = âujÎ´iâ2L2(â¦Î´N )+ âujÎ´iâ2L2(â¦Î´i\u00E2¦Î´N ) is uniformly boundedindependent of i. The first term is bounded by (4.2.11) and the second termis bounded by (4.2.4) of Lemma 4.2.2. This together with (4.2.10) shows thatâu jÎ´iâH1(â¦Î´i ) is uniformly bounded. Thus uj â H1(â¦\u00CE£).For any function Ï âW 1,2â©Lâ(â¦), define ÏÎ´ = ÏÏÎ´ where ÏÎ´ is defined inLemma 4.2.3. Since u j is a harmonic function on â¦ \u00CE£ and satisfies âu jâÎ· = Ï ju j68on ââ¦\u00E2Î£, and ÏÎ´ vanishes near Σ, we haveâ«â¦\u00CE£âu j Â·âÏÎ´ = Ï jâ«ââ¦\u00E2Î£u jÏÎ´ ,and so â«â¦Ïâu j Â·âÏÎ´ +ÏÎ´âu j Â·âÏ = Ï jâ«ââ¦u jÏÎ´ . (4.2.12)From Hölderâs inequality it follows thatâ£â£â£â£â«â¦Ïâu j Â·âÏÎ´â£â£â£â£â¤ â«â¦ |Ïâu j||âÏÎ´ | â¤(â«â¦|Ïâu j|2) 12(â«â¦|âÏÎ´ |2) 12.Therefore, by Lemma 4.2.3â«â¦Ïâu j Â·âÏÎ´ â 0 as δ â 0.Since |ÏÎ´ | â¤ |Ï| â Lâ(â¦) and u j âH1(â¦\u00CE£), taking limits both sides of (4.2.12)and applying the dominated convergence theorem, we obtainâ«â¦âu j Â·âÏ = Ï jâ«ââ¦u jÏwhich implies the claim.Claim 3. The set of functions {u j} is orthonormal on the boundary of â¦, i.e.,â«ââ¦u jul = δ jl for j, l â¥ 0.Proof. Let ε > 0. Since u j are Steklov eigenfunctions by Claim 2, and there-fore smooth by the regularity theory, there exists sufficiently small s > 0 such that|â«Lsâ©ââ¦ u jul| < ε/4. According to Hölderâs inequality and (4.2.3) and (4.2.5)of Lemma 2.2, we may furthermore assume that s is chosen sufficiently smallsuch that |â«TÎ´i u jδiulδi |< ε/4 and |â«(LsLÎ´i )â©ââ¦ u jδiulδi |< ε/4 hold for i large enough.The fact that u jδi uniformly converges to uj in any compact subset implies that69|â«ââ¦Ls u jÎ´iulÎ´iâ â«ââ¦Ls u jul|< ε/4 for i large enough. Hence it follows thatâ£â£â£â£â«ââ¦ u julâÎ´ jlâ£â£â£â£< â£â£â£â£â«ââ¦Ls u julâÎ´ jlâ£â£â£â£+ Îµ/4<â£â£â£â£â«ââ¦Ls u jÎ´iulÎ´iâÎ´ jlâ£â£â£â£+ Îµ/2=â£â£â£â£â£â«ââ¦Î´iu jÎ´iulÎ´iââ«(LsLÎ´i )â©ââ¦u jÎ´iulÎ´iââ«TÎ´iu jÎ´iulÎ´iâÎ´ jlâ£â£â£â£â£+ Îµ/2<â£â£â£â£â£â«(LsLÎ´i )â©ââ¦u jÎ´iulÎ´iâ£â£â£â£â£+â£â£â£â£â£â«TÎ´iu jÎ´iulÎ´iâ£â£â£â£â£+ ε/2< ε/4+ ε/4+ ε/2 = ε.Since ε > 0 is arbitrary this yields the desired result.Theorem 4.1.1 is proved if we can show that Ï j = Ï j(â¦). This is what we shalldo in the following, by applying induction. It is easy to see that limÎ´â0Ï0(â¦Î´ ) =Ï0(â¦) since they are always zero. Hereafter, we fix j and assume that for each0 â¤ i â¤ jâ1, ui is a Steklov eigenfunction of â¦ with eigenvalue Ïi(â¦). First, weshow that Ï j(â¦)â¤ Ï j. To see this, from Claim 3 it follows thatâ«ââ¦u j =â«ââ¦u1u j = · · Â·=â«ââ¦u jâ1u j = 0,which guarantees that u j is an admissible test function for Ï j(â¦), and thus Ï j â¥Ï j(â¦). To prove the theorem, we need to prove the reverse inequality.Denote the j-th Steklov eigenfunction of â¦ by v, and definef = vâjâ1âi=0(â«ââ¦Î´vuiÎ´)uiÎ´ .In fact, f is the projection of v to the orthogonal complement of first j eigenspacesof â¦Î´ . Therefore, â«ââ¦Î´f uiδ = 0, for i = 0, · · · , jâ1.Thus f is an admissible test function for Ï j(â¦Î´ ).70On the one hand, letâs estimate the denominator of the Rayleigh quotient,â«ââ¦Î´f 2 =â«ââ¦Î´v2âjâ1âi=0(â«ââ¦Î´vuiÎ´)2. (4.2.13)Itâs clear thatlimÎ´â0â«ââ¦Î´v2 =â«ââ¦v2andlimÎ´â0â«ââ¦Î´vuiÎ´ =â«ââ¦vui = 0 for i = 0, · · · , jâ1.On the other hand, we may estimate the numerator of the Rayleigh quotient asfollowsâ«â¦Î´|â f |2 =â«â¦Î´|âv|2+jâ1âi=1(â«ââ¦Î´vuiÎ´)2 â«â¦Î´|âuiÎ´ |2â2jâ1âi=1(â«ââ¦Î´vuiÎ´)â«â¦Î´ãâv,âuiÎ´ ã.(4.2.14)Similarly, â«â¦Î´|âuiÎ´ |2 = Ï1(â¦Î´ )<C1and â«â¦Î´ãâv,âuiÎ´ ã â¤C2(â«â¦Î´|âuiÎ´ |2) 12â¤C3,where all the constant C1,C2,C3 are independent of δ for small δ .In the end, we combine the estimates (4.2.13), (4.2.14) in the characterization(4.2.1) and take the limit to getlimsupÎ´â0Ï j(â¦Î´ )â¤ limÎ´â0â«â¦Î´ |â f |2â«ââ¦Î´ f2 =â«â¦ |âv|2â«ââ¦ v2= Ï j(â¦).This completes the proof of the theorem.714.3 Some Results on Shape Optimization for SteklovEigenvalues in RnFor any bounded domainâ¦âRn there are upper bounds on all normalized Stekloveigenvalues, Ï j(â¦)|ââ¦| 1nâ1 â¤C(n) j 2n ([21]). An explicit upper bound for the firstnormalized Steklov eigenvalue was given in [31, Proposition 2.1], however theupper bound is not expected to be sharp. It is an open question to determine thesharp upper bound.Question 4.3.1 ([35]). On which domain (or in the limit of which sequence ofdomains) is the supremum of ÏÂ¯ j(â¦) over all bounded domains â¦â Rn realized?A consequence of Theorem 4.1.1 is that the supremum of the j-th normalizedeigenvalue among all domains is the same as the supremum among domains hav-ing relatively simple topology; that is, among domains which can be obtained byperforming surgeries up to codimension two (as discussed in Section 4.2).Another immediate consequence of our surgery result is that when n â¥ 3 theunit ball is not the maximizer for ÏÂ¯ j(â¦), even among contractible domains. Thiswas proved for j = 1 in [31, Theorem 1.1].Corollary 4.1.2. Let nâ¥ 3. For any kâ¥ 1, there exist a contractible domain â¦â inRn such thatÏÂ¯ j(â¦â)> ÏÂ¯ j(Bn)for j = 1, · · · ,k.Proof. Let Bnε denote the ball of radius ε centred at the origin in Rn. It was shownin [31, Proposition 3.1] that for ε > 0 sufficiently small, ÏÂ¯ j(Bn Bnε)> ÏÂ¯ j(Bn). Wemay then perform a 1-dimensional surgery on Bn Bnε to construct a contractibledomain, while changing the Steklov eigenvalues by an arbitrarily small amount.For example, let Bnε,δ denote the domain obtained by removing a δ -tube around aradial segment (see Figure 4.1). Then since n â¥ 3, by Theorem 4.1.1, for suffi-ciently small ε > 0 we havelimÎ´â0ÏÂ¯ j(Bnε,δ ) = ÏÂ¯ j(Bn Bnε)> ÏÂ¯ j(Bn).72Figure 4.1: The construction of the contractible domain Bnε,δ in Rn for nâ¥ 3:δ is the radius of the neck and ε is the radius of the inner ball, and wedelete the neck and inner ball from the unit ball.Therefore, for any fixed k, we can find a small δ > 0 such that ÏÂ¯ j(Bnε,δ )> ÏÂ¯ j(Bn)for j = 1, · · · ,k. We let â¦â = Bnε,δ .As discussed in the introduction, this result is not surprising, especially in lightof [34] which shows that the supremum of the j-th normalized eigenvalue amongsimply connected domains in R2 is achieved in the limit by a sequence of simplyconnected domains degenerating to the disjoint union of j identical discs. We nowconsider contractible domains of this type in higher dimensions.Consider the necklace-like contractible domain â¦nÎµ,l in Rn which is the unionof l n-dimensional unit balls centred along a common axis and positioned such thatadjacent balls overlap by a distance ε2 along the axis; see Figure 4.2. As ε tendsto zero, â¦nÎµ,l converges to l identical unit balls touching tangentially at lâ1 points,which we denote by p1, · · · , plâ1 (see Figure 4.3). We denote l disjoint unit ballsby unionsqlBn.Using similar ideas as in the proof of Theorem 4.1.1, we show that the Steklovspectrum of â¦nÎµ,l converges to that of unionsqlBn as ε â 0.73r = 1Figure 4.2: The construction of â¦nÎµ,l in Rn, nâ¥ 2: l overlapping unit balls.p1 p2 plâ1r = 1Figure 4.3: Limit of â¦nÎµ,l in Rn, nâ¥ 2, as ε â 0.Theorem 4.3.2. For every j â¥ 0, the j-th normalized Steklov eigenvalue of â¦nÎµ,l ,where n â¥ 2, converges to the j-th normalized Steklov eigenvalue of the disjointunion of l unit balls as ε tends to zero, i.e.,limÎµâ0+ÏÂ¯ j(â¦nÎµ,l) = Ï j(unionsqlBn) · (l · |Snâ1|)1nâ1 . (4.3.1)Remark 4.3.3. (1) The above theorem holds for more general overlapping do-mains. For example, in the construction we could replace the unit ball by thedomain Bnε,δ from the proof of Corollary 4.1.2 (Figure 4.1). More generally, thesame proof works for domains with smooth boundary overlapping at elliptic pointson the boundary, and such that other intersections do not occur; (2) Comparing(4.3.1) with Theorem 1.3.1 in [34], the index of the Steklov eigenvalue doesnât haveto be same as the number l of balls we overlap. (3) This result was proved in [14,Example 3]. However, we use a different approach here.Proof of Theorem 4.3.2. Since |ââ¦nÎµ,l| converges to l|Snâ1|, it suffices to prove that74limÎµâ0+Ï j(â¦nÎµ,l) = Ï j(unionsqlBn). (4.3.2)We fix l â¥ 2 and nâ¥ 2, and without ambiguity, we simply use the notation â¦Îµ forâ¦nÎµ,l in the proof. We split the proof into four steps.Step 1: Uniform energy bound and uniform convergence in compact sub-sets. We consider the Steklov eigenvalue problem onâ¦Îµ . Let {u jÎµ}âj=0 be L2(ââ¦Îµ)-orthonormal Steklov eigenfunctions of â¦Îµ such that u jε is an eigenfunction witheigenvalue Ï j(â¦Îµ). By an argument similar to the proof of Claim 1, for each fixedj â N, there exists ε0 > 0 such that for all ε < ε0, Ï j(â¦Îµ), and hence the energy,is uniformly bounded independent of ε . As in the proof of Theorem 4.1.1, ellipticboundary estimates give uniform bounds on u jε and its derivatives up to ââ¦Îµ , andthere exists a sequence Îµiâ 0 such that for each j, u jεi converges in C2(K) as iââon any compact subset K of unionsqlBn {p1, . . . , plâ1} to a harmonic function u j onunionsqlBn satisfyingâu jâÎ·= Ï ju j on unionsql âBn {p1, . . . , plâ1}with Ï j = limiââÏ j(â¦Îµi).Step 2: Nonconcentration of L2 norms of eigenfunctions around p1, · · · , plâ1.For a point p on the boundary of the unit ball Bn, we consider the geodesic ballBs(p) inBn centred at p with radius s. Notice that if sâ¥ ε and p is chosen along theaxis of the domainâ¦Îµ thenBs(p)will contain the intersection of the adjacent over-lapping balls. The boundary ofBs(p) consists of two parts: Î1s (p) = âBs(p)â©Band Î2s (p) = âBs(p)\u00CE2s (p). In the following we choose r0 small such thatBs(p)is uniformly equivalent to a Euclidean half ball of radius s for s < r0.We claim that for any smooth function u onBs(p) with u = 0 on Î1s (p),â«Bs(p)u2 â¤C1(n)s2â«Bs(p)|âu|2 (4.3.3)and â«Î2s (p)u2 â¤C2(n)sâ«Bs(p)|âu|2. (4.3.4)Here C1(n),C2(n) are constants depending only on the dimension n. The above twoPoincare inequalities follow from estimates of the first Dirichlet-Neumann eigen-75value and the first Dirichlet-Steklov eigenvalue of a Euclidean half ball (or see [32,Lemma 4.5 & Lemma 4.6]).According to [32, Lemma 4.3] there exists s1 and a smooth cut-off function ζ ,which is 0 onBr0 Bs1 and 1 onBs, such that for any smooth function u definedonBr0 , â«Br1Bs|âÎ¶ |2u2 â¤C(s)(â«Br0Bsu2+â«Br0|âu|2). (4.3.5)Here s < s1 = s1(s) < r0 and s1(s) = o(1),C(s) = o(1). Now if u is any smoothfunction defined on the unit ball Bn, applying ζu to (4.3.3) we haveâ«Bs(p)u2 â¤â«Bs1 (p)(Î¶u)2â¤C1(n)s21â«Bs1 (p)|âÎ¶u|2â¤C1(n)s21(â«Bs1 (p)|âu|2+â«Bs1 (p)Bs(p)|âÎ¶ |2u2)â¤C1(s)(â«Br0 (p)|âu|2+â«Br0 (p)Bs(p)u2)(4.3.6)where C1(s) = o(1), and we used (4.3.5) in the last inequality. Similarly, applyingζu to (4.3.4) we obtainâ«Î2s (p)u2 â¤C2(s)(â«Br0 (p)|âu|2+â«Br0 (p)Bs(p)u2)(4.3.7)where C2(s) = o(1).Now we consider the eigenfunction u jε on domain â¦Îµ . In what follows we shallshow that interior and boundary L2 norms of u jε donât concentrate near p1, . . . , plâ1as ε â 0. The common axis of the domain â¦Îµ intersects the boundaries of theoverlapping balls at 2lâ 2 points, p(1)1 (ε), p(2)1 (ε), · · · , p(1)lâ1(Îµ), p(2)lâ1(Îµ), such thatas ε â 0, p(1)k (ε) and p(2)k (ε) converge to pk for k = 1, · · · , lâ 1. We apply theinequality (4.3.6) at these points to the restriction of the function u jε to each ball.76Summing the inequalities yields that for any ε < s < s1(s)< r0,â«âª2i=1âªlâ1k=1Bs(p(i)k (ε))(u jε)2 â¤ 2C1(s)(â«â¦Îµ|âu jε |2+â«âª2i=1âªlâ1k=1Br0 (p(i)k (ε))Bs(p(i)k (ε))(u jÎµ)2)â¤C Â·C1(s)where the last inequality follows from the trace inequality and the uniform energybound. Similarly, applying (4.3.7) we obtain that for any ε < s < s1(s)< r0,â«âª2i=1âªlâ1k=1Î2s (p(i)k (ε))\u00CE2ε (p(i)k (ε))(u jε)2 â¤â«âª2i=1âªlâ1k=1Î2s (p(i)k (ε))(u jÎµ)2â¤ 2C2(s)â«â¦Îµ|âu jε |2+â«âª2i=1âªlâ1k=1Br0 (p(i)k (ε))Bs(p(i)k (ε))(u jÎµ)2â¤C Â·C2(s).Hence following the ideas of the proofs of Claims 2 and 3, âu jÎµâH1(â¦Îµ ) is uni-formly bounded and therefore u j â H1(unionsqlBn), u j extends to a Steklov eigenfunc-tion of unionsqlBn, and {u j} are orthonormal on the boundary of unionsqlBn.Step 3: Construction of cut-off function. Define the following cut-off func-tion on unionsqlBn,ÏÎµ =ï£±ï£´ï£´ï£´ï£²ï£´ï£´ï£´ï£³0, r < Îµ2lnrâlnÎµ2lnÎµâlnÎµ2 ε2 â¤ r â¤ ε1 r > εwhere r is the distance function to the nearest of the points p1, . . . , plâ1. Let Tt ={x â unionsqlBn : r(x)â¤ t}. We haveâ«unionsqlBn|âÏÎµ |2 = 1(lnÎµ)2â«TÎµTÎµ21r2â¤ (2lâ2)C(n)(lnÎµ)2â« ÎµÎµ2rnâ3 dr =C(n, l)εn(ε)(4.3.8)where ε2(ε) =â1/ lnε and εn(ε) = Îµnâ2(1â Îµnâ2)/(lnÎµ)2 for nâ¥ 3.Step 4: Proof of the convergence of eigenvalues. We now use induction toprove that (4.3.2) holds: limÎµâ0Ï j(â¦Îµ) = Ï j(unionsqlBn). It is clear that Ï0(â¦Îµ) = 0 for77any ε and Ï0(unionsqlBn) = 0, and so (4.3.2) holds for j = 0. For fixed j â N, supposethat (4.3.2) holds for i= 0,1, · · · , jâ1; that is, ui is a Steklov eigenfunction of unionsqlBnwith eigenvalue Ïi(unionsqlBn) for i = 0,1, · · · , jâ1. According to Step 1, we haveâ«â (unionsqlBn)u j =â«â (unionsqlBn)u1u j = · · Â·=â«â (unionsqlBn)u jâ1u j = 0.Then u j is an admissible test function for the j-th eigenvalue of unionsqlBn. ThusÏ j(unionsqlBn)â¤ Ï j, and we haveliminfÎµâ0+Ï j(â¦Îµ)â¥ Ï j(unionsqlBn). (4.3.9)In order to prove the theorem, we need to prove the reverse inequality. We willuse a j-th eigenfunction of unionsqlBn, which we denote by v, to construct an admissibletest function for the j-th eigenvalue of â¦Îµ2 , where ε2 is the square of ε . This partof the proof is slightly different from the proof of Theorem 4.1.1, since we need tocut off and glue functions to construct the admissible test function to approximatethe j-th eigenvalue of â¦Îµ2 .Let f = vÏÎµ ââ jâ1i=0 (â«ââ¦Îµ2vÏÎµuiÎµ2)uiÎµ2 , where we recall that {uiÎµ2}âi=0 are or-thonormal eigenfunctions of â¦Îµ2 . By construction, f is orthogonal to u0ε2 , · · · ,ujâ1Îµ2on ââ¦Îµ2 , and soÏ j(â¦Îµ2)â¤â«â¦Îµ2|â f |2â«ââ¦Îµ2f 2.A simple calculation turns the numerator toâ«â¦Îµ2|â f |2 =â«â¦Îµ2|â(vÏÎµ)|2+jâ1âi=1Ïi(â¦Îµ2)(â«ââ¦Îµ2vÏÎµuiÎµ2)2â2jâ1âi=1â«ââ¦Îµ2vÏÎµuiÎµ2â«â¦Îµ2ãâuiÎµ2 ,â(vÏÎµ)ã. (4.3.10)78The first term can be estimated asâ«â¦Îµ2|â(vÏÎµ)|2 â¤â«â¦Îµ|âv|2+2â«TÎµTÎµ2Ï2Îµ |âv|2+ v2|âÏÎµ |2â¤â«unionsqlBn|âv|2+C1|TÎµ Tε2 |+C2â«TÎµTÎµ2|âÏÎµ |2=â«unionsqlBn|âv|2+C3(Îµ) (4.3.11)with C3(Îµ)â 0 as ε â 0+ by (4.3.8). Itâs not hard to check that for 0â¤ iâ¤ j,limÎµâ0+â«ââ¦Îµ2vÏÎµuiÎµ2 =â«unionsqlBnvui = 0and â£â£â£â£â«â¦Îµ2 ãâuiÎµ2 ,â(vÏÎµ)ãâ£â£â£â£â¤ (Ïi(â¦Îµ2)) 12 (â«â¦Îµ2 |â(vÏÎµ)|2) 12.Since Ïi(â¦Îµ2) are uniformly bounded in ε , and by (4.3.11), the last two terms ofthe right hand side of (4.3.10) both tend to zero as ε â 0+. In all, we have aninequality for the numerator,â«â¦Îµ2|â f |2 â¤â«unionsqlBn|âv|2+C4(Îµ)with C4(Îµ)â 0 as ε â 0+.On the other hand, we expand the denominator as followsâ«ââ¦Îµ2f 2 =â«ââ¦Îµ2(vÏÎµ)2âjâ1âi=0(â«ââ¦Îµ2vÏÎµuiÎµ2)2ââ«â (unionsqlBn)v2as ε â 0, since the second term tends to zero as ε â 0.Now we combine all the estimates to getlimsupÎµâ0Ï j(â¦Îµ2)â¤ limÎµâ0â«â¦Îµ2|â f |2â«ââ¦Îµ2f 2â¤â«unionsqlBn |âv|2â«â (unionsqlBn) v2 = Ï j(unionsqlBn).Therefore together with (4.3.9) we have limÎµâ0Ï j(â¦Îµ) = Ï j(unionsqlBn).79As a special case of Theorem 4.3.2, we obtain the following generalization of[34, Theorem 1.3.1] to higher dimensions.Theorem 4.3.4. For the domains â¦nÎµ, j â Rn (see Figure 4.2), nâ¥ 2, we havelimÎµâ0+ÏÂ¯ j(â¦nÎµ, j) = ( j · |Snâ1|)1nâ1 .Proof. When l = j, Ï j(unionsq jBn) = 1, thus limÎµâ0+ Ï j(â¦nÎµ, j) = ( j · |Snâ1|)1nâ1 .Following Remark 4.3.3, if we take the domains that we overlap to be Bnε,δ ,we see that when n â¥ 3 the domains â¦nÎµ, j do not attain the supremum of the j-th Steklov eigenvalue in the limit as ε â 0 for any j, in contrast to the case indimension two [34].Theorem 4.1.3. For nâ¥ 3 the supremum of the j-th normalized Steklov eigenvalueamong contractible domains in Rn is not achieved in the limit by a sequence ofcontractible domains degenerating to the disjoint union of j identical round balls.Proof. Let â¦ËnÎµ, j be the domain obtained by overlapping j copies of the domainBnε,δ . Then by the proof of Theorem 4.3.2,limÎµâ0+ÏÂ¯ j(â¦ËnÎµ, j) = j1nâ1 Â·Ï j(Bnε,δ ) · |âBnÎµ,Î´ |1nâ1 .By Theorem 4.3.4, Corollary 4.1.2, and Theorem 4.1.1 it follows thatlimÎµâ0+ÏÂ¯ j(â¦ËnÎµ, j)> limÎµâ0+ÏÂ¯ j(â¦nÎµ, j).We end this section with following corollary.Corollary 4.3.5. For certain j and n, if ε > 0 is sufficiently small, then ÏÂ¯ j(â¦nÎµ, j)>ÏÂ¯ j(Bn).Proof. By Theorem 4.3.4, we have liminfÎµâ0+ ÏÂ¯ j(â¦nÎµ, j) = ( jnÏn)1nâ1 > ÏÂ¯ j(Bn) ifj1/(nâ1) > Ï j(Bn). By simple calculation, this only holds in certain cases. Forexample, when n= 3, it holds for all j except when j is 4,9,16, . . .. For higher n, it80holds all j less than a certain finite number depending on n. In these cases we canchoose sufficiently small ε such that ÏÂ¯ j(â¦nÎµ, j)> ÏÂ¯ j(Bn).81Bibliography[1] R. Adams and J. Fournier. Sobolev spaces, volume 140. Elsevier/AcademicPress, second edition, 2003. â page 68[2] S. Agmon, A. Douglis, and L. Nirenberg. Estimates near the boundary forsolutions of elliptic partial differential equations satisfying general boundaryconditions. I. Comm. Pure Appl. Math., 12:623â727, 1959. â page 44[3] N. S. Aiex and H. Hong. Compactness of the space of free boundary cmcsurfaces with bounded topology. arXiv:2012.00956. â page v[4] N. S. Aiex and H. Hong. Index Estimates for Surfaces with Constant MeanCurvature in 3-dimensional Manifolds. Calc. Var. Partial DifferentialEquations, 60(3), 2021. â pages v, 39[5] W. K. Allard. On the first variation of a varifold. Ann. of Math. (2), 95:417â491, 1972. â page 54[6] W. K. Allard. On the first variation of a varifold: boundary behavior. Ann. ofMath. (2), 101:418â446, 1975. â page 54[7] L. Ambrozio, A. Carlotto, and B. Sharp. Comparing the Morse index andthe first Betti number of minimal hypersurfaces. J. Differential Geom., 108(3):379â410, 2018. â pages 11, 12, 14, 18, 24, 35[8] L. Ambrozio, A. Carlotto, and B. Sharp. Index estimates for free boundaryminimal hypersurfaces. Math. Ann., 370(3-4):1063â1078, 2018. â pages11, 12, 18[9] L. Ambrozio, A. Carlotto, and B. Sharp. Compactness analysis for freeboundary minimal hypersurfaces. Calc. Var. Partial Differential Equations,57(1):Paper No. 22, 39, 2018. â pages 40, 47, 48, 49, 55, 56, 5782[10] J. L. Barbosa and P. Bérard. Eigenvalue and âtwistedâ eigenvalue problems,applications to CMC surfaces. J. Math. Pures Appl. (9), 79(5):427â450,2000. â pages 3, 14, 17, 18[11] J. L. Barbosa and M. do Carmo. Stability of hypersurfaces with constantmean curvature. Math. Z., 185(3):339â353, 1984. â page 10[12] J. L. Barbosa, M. do Carmo, and J. Eschenburg. Stability of hypersurfacesof constant mean curvature in Riemannian manifolds. Math. Z., 197(1):123â138, 1988. â pages 10, 29[13] D. Bucur, V. Ferone, C. Nitsch, and C. Trombetti. Weinstock inequality inhigher dimensions. arXiv:1710.04587 (To appear in J. Differential Geom),2019. â pages 8, 58[14] D. Bucur, A. Giacomini, and P. Trebeschi. Lâ bounds of stekloveigenfunctions and spectrum stability and domain variation. PreprintCVGMT, 2019. â page 74[15] M. P. Cavalcante and D. F. de Oliveira. Index Estimates for Free BoundaryConstant Mean Curvature Surfaces. Pacific Journal of Mathematics, 305(1):1â14, 2018. â pages 3, 14, 36[16] M. P. Cavalcante and D. F. de Oliveira. Lower bounds for the index ofcompact constant mean curvature surfaces in R3 and S3. Rev. Mat.Iberoamericana, 36(1):195â206, 2020. â pages 3, 14, 29[17] O. Chodosh and D. Maximo. On the topology and index of minimalsurfaces. J. Differential Geom., 104(3):399â418, 2016. â page 11[18] O. Chodosh and D. Maximo. On the topology and index of minimal surfacesII. arXiv:1808.06572 (To appear in J. Differential Geom), 2018. â page 11[19] H. I. Choi and R. Schoen. The space of minimal embeddings of a surfaceinto a three-dimensional manifold of positive ricci curvature. Invent. Math,81(3):387â394, 1985. â page 5[20] F. Codá Marques. Minimal surfaces: variational theory and applications. InProceedings of the International Congress of MathematiciansâSeoul 2014.Vol. 1, pages 283â310. Kyung Moon Sa, Seoul, 2014. â page 14[21] B. Colbois, A. El Soufi, and A. Girouard. Isoperimetric control of thesteklov spectrum. J. Funct. Anal, 261(5):1384â1399, 2011. â page 7283[22] T. H. Colding and W. P. Minicozzi, II. Embedded minimal surfaces withoutarea bounds in 3-manifolds. In Geometry and topology: Aarhus (1998),volume 258 of Contemp. Math., pages 107â120. Amer. Math. Soc.,Providence, RI, 2000. â page 40[23] M. do Carmo and C. K. Peng. Stable complete minimal hypersurfaces. InProceedings of the 1980 Beijing Symposium on Differential Geometry andDifferential Equations, Vol. 1, 2, 3 (Beijing, 1980), pages 1349â1358. Sci.Press Beijing, Beijing, 1982. â page 11[24] X. Fan, L. Tam, and C. Yu. Extremal problems for steklov eigenvalues onannuli. Calc. Var. Partial Differ. Equ, 1(54):1043â1059, 2017. â page 2[25] D. Fischer-Colbrie and R. Schoen. The structure of complete stable minimalsurfaces in 3-manifolds of nonnegative scalar curvature. Comm. Pure Appl.Math., 33(2):199â211, 1980. â page 11[26] A. Fraser and M. M.-c. Li. Compactness of the space of embedded minimalsurfaces with free boundary in three-manifolds with nonnegative Riccicurvature and convex boundary. J. Differential Geom., 96(2):183â200, 2014.â pages 5, 6, 38, 40[27] A. Fraser and P. Sargent. Existence and classification of S1 invariant freeboundary minimal annuli and mobius bands in B. J. Geom. Anal., 1(2):1â23,2020. â page 2[28] A. Fraser and R. Schoen. The first steklov eigenvalue, conformal geometry,and minimal surfaces. Adv. Math., 226(5):4011â4030, 2011. â page 2[29] A. Fraser and R. Schoen. Minimal surfaces and eigenvalue problems. Geom.Anal. Math. Relativ. Nonlinear Partial Differ. Equ., (599):105â121, 2013. âpage 7[30] A. Fraser and R. Schoen. Sharp eigenvalue bounds and minimal surfaces inthe ball. Inventiones Mathematicae, 203(3):823â890, 2016. â pages 2, 7, 59[31] A. Fraser and R. Schoen. Shape optimization for the steklov problem inhigher dimensions. Adv. Math., 348:146â162, 2019. â pages8, 58, 59, 62, 72[32] A. Fraser and R. Schoen. Some results on higher eigenvalue optimization.Calc. Var. Partial Differential Equations, 59(151):1â22, 2020. â pages60, 7684[33] D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations ofsecond order, volume 224 of Grundlehren der MathematischenWissenschaften [Fundamental Principles of Mathematical Sciences].Springer-Verlag, Berlin, second edition, 1983. â pages 43, 68[34] A. Girouard and I. Polterovich. On the hersch-payne-schiffer inequalities forsteklov eigenvalues. Funct. Anal. Appl., 44:106â117, 2010. â pages9, 60, 73, 74, 80[35] A. Girouard and I. Polterovich. Spectral geometry of the steklov problem. J.Spectral Theory, 2016. â page 72[36] M. Grüter and J. Jost. Allard type regularity results for varifolds with freeboundaries. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 13(1):129â169, 1986.â page 54[37] L. Hauswirth, J. Pérez, P. Romon, and A. Ros. The periodic isoperimetricproblem. Trans. Amer. Math. Soc., 356(5):2025â2047, 2004. â page 32[38] J. Hersch, L. E. Payne, and M. Schiffer. Some inequalities for stekloffeigenvalues. Arch. Rational Mech. Anal, 57:99â114, 1975. â page 60[39] H. Hong. Higher dimensional surgery and steklov eigenvalues.arXiv:2007.06734, 2021. â page v[40] A. Jankowski and R. Rubinsztein. Functions with non-degenerate criticalpoints on manifolds with boundary. Comment. Math. Prace Mat., 16:99â112, 1972. â page 50[41] B. E. Kantor and S. A. Frangulov. Isometric immersions of two-dimensionalriemann manifolds in pseudo-euclidean space. Mathematical notes of theAcademy of Sciences of the USSR, 36(3):721â725, Sep 1984. â page 31[42] H. Matthiesen and R. Petrides. Ofree boundary minimal surfaces of anytopological type in euclidean balls via shape optimization.arXiv:2004.06051. â page 59[43] A. Mendes. Rigidity of free boundary surfaces in compact 3-manifolds withstrictly convex boundary. J. Geom. Anal., 28(2):1245â1257, 2018. â page39[44] A. Neves. New applications of min-max theory. In Proceedings of theInternational Congress of MathematiciansâSeoul 2014. Vol. II, pages939â957. Kyung Moon Sa, Seoul, 2014. â pages 4, 1485[45] I. Nunes. On stable constant mean curvature surfaces with free boundary.Math. Z., 287(1-2):473â479, 2017. â page 11[46] A. V. Pogorelov. On the stability of minimal surfaces. Dokl. Akad. NaukSSSR, 260(2):293â295, 1981. â page 11[47] A. Ros. One-sided complete stable minimal surfaces. J. Differential Geom.,74(1):69â92, 2006. â pages 3, 11, 18[48] A. Ros and E. Vergasta. Stability for hypersurfaces of constant meancurvature with free boundary. Geom. Dedicata, 56(1):19â33, 1995. â pages11, 17[49] P. Sargent. Index bounds for free boundary minimal surfaces of convexbodies. Proc. Amer. Math. Soc., 145(6):2467â2480, 2017. â pages 11, 16[50] A. Savo. Index bounds for minimal hypersurfaces of the sphere. IndianaUniv. Math. J., 59(3):823â837, 2010. â page 11[51] B. Sharp. Compactness of minimal hypersurfaces with bounded index. J.Differential Geom., 106(2):317â339, 2017. â pages 40, 44, 55[52] L. Simon. A strict maximum principle for area minimizing hypersurfaces. J.Differential Geom., 26(2):327â335, 1987. â page 40[53] R. Souam. On stability of stationary hypersurfaces for the partitioningproblem for balls in space forms. Math. Z., 224(2):195â208, 1997. â page10[54] R. Souam. On stable constant mean curvature surfaces in S2ÃR andH2ÃR. Trans. Amer. Math. Soc., 362(6):2845â2857, 2010. â pages 29, 30[55] A. Sun. Compactness of constant mean curvature surfaces in athree-manifold with positive Ricci curvature. Pacific J. Math., 305(2):735â756, 2020. â pages 5, 41[56] F. Torralbo and F. Urbano. Compact stable constant mean curvature surfacesin homogeneous 3-manifolds. Indiana Univ. Math. J., 61(3):1129â1156,2012. â pages 11, 34, 35[57] R. Weinstock. Inequalities for a classical eigenvalue problem. J. RationalMech. Anal., 3:745â753, 1954. â pages 8, 58, 60[58] B. White. Curvature estimates and compactness theorems in 3-manifolds forsurfaces that are stationary for parametric elliptic functionals. Invent. Math.,88(2):243â256, 1987. â pages 40, 45, 4786
Thesis/Dissertation
eng
Mathematics
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Chen, Jingyi
Fraser, Ailana
Index estimates and compactness for constant mean curvature surfaces and Steklov eigenvalues
Text