Minimal Hypersurfaces of the RoundSpherebyPamela SargentB.Sc., Mount Allison University, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2013c? Pamela Sargent 2013AbstractThe purpose of this thesis is to discuss a conjectured classification concerningthe index of non-totally geodesic minimal hypersurfaces of the n-dimensionalstandard sphere of radius one Sn. We briefly discuss the basic theory of min-imal submanifolds before turning our attention to minimal submanifolds andhypersurfaces in Sn. We present some results of Simons which show thatany minimal submanifold of Sn is unstable, and how the totally geodesicSk ? Sn are characterized by their index. We then present a related conjec-ture which claims that the Clifford hypersurfaces are also characterized bytheir index in a similar way, discuss the most recent developments relatedto the conjecture, and give Urbano?s proof of the conjecture for the specialcase when n = 3.iiPrefaceThe topic of this thesis was jointly chosen by the author and her supervisor,Dr. Ailana Fraser.This thesis surveys a collection of known results. While the content isnot original, the organization and presentation, to the best of the author?sknowledge, are. Moreover, the author has provided a few modifications inthe proof of the first main result and many details in the proof of the secondmain result that are not presented in the original proof.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . vi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Some Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Notation and Conventions . . . . . . . . . . . . . . . 21.1.2 Riemannian Submanifolds . . . . . . . . . . . . . . . 31.1.3 Minimal Submanifolds . . . . . . . . . . . . . . . . . 51.2 Operators on Riemannian Manifolds . . . . . . . . . . . . . . 61.2.1 Basic Differential Operators on Riemannian Manifolds 71.2.2 The Jacobi Operator and the Morse Index . . . . . . 81.2.3 Spectral Theory of Strongly Elliptic Operators . . . . 102 Classification Results for Minimal Submanifolds of Sn . . 132.1 Minimal Submanifolds of Sn . . . . . . . . . . . . . . . . . . 132.1.1 Isometric and Minimal Immersions into Sn . . . . . . 132.1.2 Simons Classification Result . . . . . . . . . . . . . . 162.2 Non-totally Geodesic Minimal Hypersurfaces of Sn . . . . . . 252.2.1 A Classification Conjecture . . . . . . . . . . . . . . . 252.2.2 Urbano?s Result: Proof of the Conjecture when n = 3 283 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44ivTable of ContentsBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46vAcknowledgementsI would like to thank my supervisor, Dr. Ailana Fraser, who introduced meto this field. I have benefited greatly from her expertise, encouragement andpatience, and this thesis would not have been achieved without her.viChapter 1IntroductionThis thesis focuses on the study of minimal submanifolds, a central topic ingeometric analysis. Minimal submanifolds are critical points of the volumefunctional. The study of minimal surfaces, 2-dimensional minimal subman-ifolds, has involved such prolific mathematicians as Lagrange and Euler, aswell as the physicist Plateau. It was Plateau who investigated the surfaceobtained by dipping a wire frame into a soap solution, a physical exampleof a minimal surface, and it was his work that inspired the famous Plateausproblem: given a simple, closed curve, is there a minimal surface whoseboundary corresponds to that curve? The general solution to this problemwas found in the 1930s independently by Rado and Douglas, and Douglaslater won the Fields medal for his work on the problem. Some other notableexamples of minimal surfaces are helicoids (the geometric shape of DNAand double-spiral staircases) and catenoids (minimal surfaces obtained byrotating catenaries about their directrices). The study of minimal surfaces,besides being important in its own right, has physical applications in fluidinterface problems and navigation problems, deep connections to fundamen-tal questions in general relativity and, in fact, played a crucial role in thecelebrated proof of the century-old Poincare conjecture.The classical case of minimal submanifolds in Rn is a subject that hasalso been studied for centuries. Another natural setting for studying min-imal surfaces is in Riemannian manifolds, and one interesting case is then-dimensional sphere Sn. Here, there is a fundamental difference from thecase of Rn: every minimal submanifold in Rn is necessarily non-compact,but in Sn there exist closed minimal submanifolds. In particular, minimalsurfaces in S3 are bountiful: in 1970, Lawson [8] proved that a closed ori-entable surface of any genus can be realized as an embedded minimal surface11.1. Some Preliminariesin S3.Here we will often deal with minimal hypersurfaces of Sn, i.e. minimalsubmanifolds of Sn of dimension n ? 1. More specifically, we will be con-cerned with problems regarding minimal submanifolds and hypersurfaces ofSn and their index. Despite their name, minimal submanifolds are criticalpoints of the volume functional which, in general, are not necessarily locallyvolume minimizing. The index of a minimal submanifold corresponds tothe index of the hessian of the volume functional, and, intuitively speak-ing, is the number of independent directions in which one can deform theminimal submanifold to decrease its volume. So, a minimal submanifold islocally volume minimizing if and only if its index is 0. We will show thatthere is a characterization of the minimal submanifolds of Sn which mini-mize the index, and present a related conjecture which claims that there isalso a characterization of the non-totally geodesic minimal hypersurfaces ofSn which minimize the index. We will state known partial results relatedto the conjecture and conclude with a proof of the conjecture in the specialcase when n = 3.1.1 Some PreliminariesThe purpose of this first chapter is to briefly introduce some preliminarydefinitions, notations and conventions, and some of the basic theory of Rie-mannian submanifolds. We conclude with the first and second variationformulas for volume and the definition of a minimal submanifold.1.1.1 Notation and ConventionsLet (M, g), (M ?, g?) be Riemannian manifolds with Levi-Civita connections?,?? respectively. By C?(M,M ?) we mean the collection of smooth func-tions f : M ? M ?, and Ck(M,M ?) will denote the collection of functionsf : M ? M ? which are k times continuously differentiable. In the specialcase that M ? = R, we will simply write C?(M) or Ck(M). Let E pi?M bea vector bundle over M . We denote the collection of smooth sections of E21.1. Some Preliminariesby ?(E), i.e.?(E) = {X ? C?(M,E) | pi ?X = idM}.We take the convention of defining the Riemannian curvature endomorphismon M , RM : ?(TM)? ?(TM)? ?(TM)? ?(TM) byRM (X,Y )Z = ?X?Y Z ??Y?XZ ??[X,Y ]Z,and the Riemannian curvature tensor (field), RM : ?(TM) ? ?(TM) ??(TM)? ?(TM)? R byRM (X,Y, Z,W ) = g (RM (X,Y )Z,W ) .Unless otherwise stated, Sn will denote the standard n-dimensionalsphere of radius one with the induced metric from Rn+1.1.1.2 Riemannian SubmanifoldsIf Mk ? ?Mn is an immersed submanifold, then for every p ? M , the tan-gential and normal projections (?)T : Tp?M ? TpM , (?)N : Tp?M ? NpMnaturally decompose the tangent space of ?M :Tp?M = TpM ?NpM for p ?M.An immersed manifold ? : M ? ?M in a Riemannian manifold ?M nat-urally inherits a Riemannian manifold structure from the ambient space: if?M has Riemannian metric g and Levi-Civita connection ?, then g = ??g isa Riemannian metric with Levi-Civita connection given by?XY = d??1((?d?(X)d?(Y ))T).Thinking of M ? ?M , we will often simply write g = g|M and ? =(?)T.Then? =(?)T+(?)N= ?+?N ,31.1. Some Preliminarieswith ?N :=(?)N.Definition 1.1.2.1. The second fundamental form of M induced by ? isthe vector-valued bilinear form A defined byA(X,Y ) =(?XY)NX,Y ? TpM.It is easy to see that the A is symmetric, i.e. A(X,Y ) = A(Y,X), sinceA(X,Y ) =(?XY)N=(?YX + [X,Y ])N(g is torsion free)= A(Y,X) + 0 ([X,Y ] ? TpM).Let p ?M and let {Ei}ki=1 be an orthonormal basis for TpM .Definition 1.1.2.2. The mean curvature vector H at p ?M is given byH =k?i=1A(Ei, Ei).Definition 1.1.2.3. The squared norm of the second fundamental form atp ?M is?A?2 =k?i,j?A(Ei, Ej)?2.Later we will need to distinguish the following special type of submani-fold.Definition 1.1.2.4. A Riemannian submanifoldM ? ?M is said to be totallygeodesic if one of the following equivalent conditions holds:(i) Every g-geodesic in M is also a g?-geodesic in ?M .(ii) The second fundamental form of M vanishes identically, i.e. A ? 0.The next propositions relate the second fundamental form of M to thecurvatures of M and ?M . The first tells us how the difference in the curvatureof the submanifold and the curvature of the ambient manifold relates to thesecond fundamental form. The second tells us how the covariant derivative of41.1. Some Preliminariesthe second fundamental form also gives us information about the curvatureendomorphism of M .Proposition 1.1.2.1. (Gauss Equation) For any X,Y, Z,W ? TpMR?M (X,Y, Z,W ) = RM (X,Y, Z,W )? ?A(X,W ), A(Y,Z)?+ ?A(X,Z), A(Y,W )?,where ??, ?? denotes the metric on ?M .Proposition 1.1.2.2. (Codazzi Equation) For any X,Y, Z ? ?(TM),(RM (X,Y )Z)N = (?XA) (Y,Z)? (?YA) (X,Z),where (?XA) (Y,Z) = ?NXA(Y,Z)?A (?XY, Z)?A (Y,?XZ).1.1.3 Minimal SubmanifoldsLet ?M be a Riemannian n-manifold and ? : M ? ?M be an immersion,where M is a compact oriented k-manifold with (possibly empty) boundary?M . Let F : I ?M ? M , I = (?1, 1), be a smooth variation of ?, i.e. Fis a smooth mapping such that(i) For each t ? I, the map ?t := F (t, ?) : M ? ?M is an immersion.(ii) ?0 := F (0, ?) = ?.(iii) For each t ? I, ?t???M = ????M .Let t be the coordinate on I and let E be the section of T (M) ? N(M)given by E = dF(??t??t=0). We will suppress the time dependence and letdV denote the volume element of the metric induced by ?t so that thevolume of M at time t, Area(t), is given byArea(t) =?MdV.Then we have51.2. Operators on Riemannian ManifoldsTheorem 1.1.3.1. (The first variation formula)dAreadt????t=0=?MdivME dV = ??M?H,E? dV. (1.1)Theorem 1.1.3.2. (The second variation formula)d2Areadt2????t=0=??M|?A(?, ?), E?|2dV+?M|?NME|2dV ??MTrM ?RM (?, E)?, E?dV.(1.2)Definition 1.1.3.1. Let ? : M ? ?M be an immersion. We say that an(immersed) manifold M is a minimal submanifold of ?M if dAreadt??t=0 = 0 forevery smooth variation of ?.Lemma 1.1.3.3. M is a minimal submanifold of ?M if and only if the meancurvature vector vanishes.Proof of 1.1.3.3: It is clear from the first variation formula (1.1) that ifH ? 0, then M is a minimal submanifold of ?M .Now suppose M is a minimal submanifold of ?M and that H(p) 6= 0 forsome p ?M . Then, by continuity, ?H? > 12?H(p)? on some neighbourhoodU ? M of p. Let ? : M ? R be a smooth bump function such that?|U ? 1, and let F be a smooth variation of ? with variation field ?H (e.g.F (t, q) = expq (t?(q)H(q))). Then we have that0 =dAreadt????t=0= ??M?H,?H? dV0 ? ?12?H(p)?2?UdV0.Since?U dV0 > 0, this gives us that ?H(p)? ? 0, a contradiction.1.2 Operators on Riemannian ManifoldsIn this chapter we discuss some specific operators on Riemannian manifolds.We begin with the definitions of some common differential operators andmake note of a few results that we will make use of later in Chapter 2.1.61.2. Operators on Riemannian ManifoldsThen we will move on to discuss a specific elliptic operator on minimalsubmanifolds which comes from the second variation formula: the Jacobioperator. We will introduce the ideas of the index and stability of a minimalsubmanifold, and outline the general spectral theory for elliptic operatorson compact Riemannian manifolds.1.2.1 Basic Differential Operators on RiemannianManifoldsDefinition 1.2.1.1. Let f ? C?(M). We define:(i) The gradient of f , grad f ? ?(TM) to be the vector field characterizedby the equationdf(X) = ?grad f,X?.(ii) The Hessian of f , Hess f , to be the symmetric (0, 2)-tensor field suchthat, for X,Y ? ?(TM)Hess f(X,Y ) = ??Xgrad f, Y ? = XY f ? (?XY )f.(iii) The Laplacian of f , ?f ? C?(M) to be the function given by?f = Tr(Hess f) =k?i=1?Ei(?Eif)? (?EiEi)f,where {Ei}ki=1 is a local orthonormal frame for TM .A simple calculation shows that the Laplacian satisfies a type of productrule:?(fg) = f?g + g?f + 2?grad f, grad g?. (1.3)We will also make use of the following Green?s formula:Lemma 1.2.1.1. Let f ? C2(M), g ? C1(M) be functions such thath(grad f) has compact support. Then?Mh?f + ?gradh, grad f? dV = 0.71.2. Operators on Riemannian ManifoldsSee [4] (pg. 6) for more details. Note that (1.3) and Lemma 1.2.1.1together show that, if grad f has compact support, then0 =?M1 ??f + ?grad f, grad 1? dV =?M?f dV. (1.4)In particular, if M is compact, then (1.4) holds for any f ? C2(M).1.2.2 The Jacobi Operator and the Morse IndexWe can alternatively write the second variation formula (1.2) asd2Areadt2????t=0= ??M?E,LE?dV. (1.5)where L is the self-adjoint Jacobi operator of the second variation whichacts on a normal vector field X to M byLX = ?NMX ?R(X) + A?(X). (1.6)Here, if {Ei}ki=1 is an orthonormal frame for TM , A? is the Simons?s operatordefined byA?(X) =k?i,j=1g(A(Ei, Ej), X)A(Ei, Ej), (1.7)?NM is the Laplacian on the normal bundle?NMX =k?i=1?NEi?NEiX ?k?i=1?N(?EiEi)TX, (1.8)andR(X) := Tr[R?M (?, X)?] =k?i=1R?M (Ei, X)Ei.If M is an orientable hypersurface of ?M , then M has a trivial normalbundle (the normal bundle has a global orthonormal frame) and the Jacobioperator simplifies to an operator on functions: because we can write anynormal vector field as a function times the unit normal vector field, we can81.2. Operators on Riemannian Manifoldsidentify normal vector fields with functions, i.e. if X = fN , then we canidentify X with f , andLf = ?Mf + ?A?2f + Ric?M (N,N)f. (1.9)We will say that ? is a (Dirichlet) eigenvalue for L on ? ?M if there isa non-trivial normal vector field X ? ?(NM) such that X|?? = 0 andLX + ?X = 0. (1.10)Definition 1.2.2.1. The Morse index (or just index ) of a compact minimalsubmanifold M , denoted ind(M), is the number of negative eigenvalues ofthe Jacobi operator L acting on the space of smooth sections of the normalbundle which vanish on the boundary (counted with multiplicity).There is a quadratic form, Q, associated to the Jacobi operator that isgiven byQ(X,X) = ??M?X,LX? dV = ??M?X,?NMX ?R(X) + A?(X)?dV,(1.11)and ind(M) is the index of Q. In the case that M is a hypersurface, thissimplifies toQ(f, f) =?M|?f |2 ? (?A?2 + Ric(N,N))f2dV. (1.12)It is sometimes useful to work with this quadratic form instead of the Jacobioperator.It follows from the second variation formula (1.2) that we could havealternatively defined the Morse index of M to be the index of M as a criticalpoint for the volume functional, and so the Morse index, in some sense,describes how stable the minimal submanifold is: it gives the number ofindependent directions in which the minimal submanifold can be deformedto make its volume decrease. Thus, a stable minimal submanifold, one whichtruly minimizes (locally) the volume functional, has index zero. If the index91.2. Operators on Riemannian Manifoldsis positive, the M is said to be unstable. We will see later that there are nostable minimal submanifolds in the standard sphere Sn.1.2.3 Spectral Theory of Strongly Elliptic OperatorsHere we give a brief overview of some standard results concerning the generaltheory for strongly elliptic operators. Recall that an elliptic differentialoperator of order m on an n-dimensional manifold M is an operator P thatin local coordinates has the formP (x,D)u =?|?|?ma?(x)D?u,and whose principal symbolPm(x, ?) =?|?|?ma?(x)??is invertible for nonzero ? ? Rn. Here D = (D1, . . . , Dn), Dj = 1i??xj and? = (?1, . . . , ?n) is a multi-index so that D? = D?11 ? ? ?D?nn . A stronglyelliptic operator is an elliptic differential operator with12(Pm(x, ?) + Pm(x, ?)?) ? C???m,where Pm(x, ?)? = Pm(x, ?)T .The Jacobi operator is known to be strongly elliptic (see [16] pg. 65) andso we will later apply some of these results to the Jacobi operator when weprove a special case of the main conjecture.If L is a strongly elliptic operator andM is compact, then the spectrum ofL is composed entirely of eigenvalues {?i}?i=1. If we arrange the eigenvaluesaccording to size?1 < ?2 < ? ? ? < ?k < ? ? ?then {?i}?i=1 are discrete and ?k ? ?. Moreover, the multiplicity of eacheigenvalue is finite. In fact, the first eigenvalue has multiplicity one andthere is an eigenfunction associated to the first eigenvalue that is strictly101.2. Operators on Riemannian Manifoldspositive.The eigenspaces Ei corresponding to ?i are mutually orthogonal withrespect to the inner product on L2(M), and??i=1Ei is dense in both L2(M)and the completion of C?(M) in L2(M).If fi is an eigenfunction corresponding to ?i, then for any function u ?W 1,2(M,R) with ?u?L2 = 1 and ?u, fi?L2 = 0 for i = 1, 2, . . . j ? 1, we havethatQ(u, u) ? ?j , (1.13)with equality holding if and only if Lu+ ?ju = 0.Sometimes bounds on the index of a minimal submanifold can be ob-tained directly from getting bounds on the number of negative eigenval-ues of L, though this is more feasible in certain situations than in oth-ers. If the Ricci curvature and squared norm of the second fundamentalform are constant, then finding the eigenvalues of the Jacobi operator turnsinto the problem of finding the eigenvalues of the Laplacian on M , a morethoroughly-studied operator. In some special cases, such as when we have(M, g) = (Sk, g0), with g0 the metric induced from Rk+1, the eigenvalues ofthe Laplacian and their multiplicities are known precisely and the index canbe calculated.Theorem 1.2.3.1. The eigenvalues of the Laplacian on (Sk, g0) are givenby?j := (j ? 1)(k + j ? 2),with multiplicitydimPj?1 ? dimPj?3 =(k + j ? 1j ? 1)?(k + j ? 3j ? 3),where Pj is the space of homogeneous polynomials of degree j on Rk+1.The proof is given by Sakai in [15] (pg. 272).The following theorem shows that the spectrum of the Laplacian on aproduct manifold (endowed with the product metric) is completely deter-mined by the spectra of the Laplacian on each of the component manifolds.111.2. Operators on Riemannian ManifoldsHere Spec(M, g) denotes the spectrum of the Laplacian on the Riemannianmanifold (M, g).Theorem 1.2.3.2. Let (M, g) and (N,h) be Riemannian manifolds. Onthe product manifold (M ?N, g ? h),Spec(M ?N, g ? h) ={?+ ??? ? ? Spec(M, g), ? ? Spec(N,h)}.The details are given by Berger in [2] (pg. 143).12Chapter 2Classification Results forMinimal Submanifolds of SnIn this chapter we specifically focus on minimal submanifolds of Sn. We be-gin by deriving a condition for which an isometric immersion ? : M ? Snis actually a minimal immersion. Following Simons?s original exposition[16], we prove that out of all k-dimensional minimal submanifolds of Sn,the standard embedded Sk ? Sn are the ones which minimize the index.We discuss a related conjecture which claims that amongst the non-totallygeodesic minimal hypersurfaces, the Clifford hypersurfaces minimize the in-dex. Finally, we present Urbano?s proof of this conjecture in the special casewhen n = 3.2.1 Minimal Submanifolds of SnHere we present some properties of minimal submanifolds of Sn.2.1.1 Isometric and Minimal Immersions into SnLater we will use the following proposition from Lawson?s book [9] (pg.15) which tells us how the Laplacian acts on the component functions ofisometric and minimal immersions into the standard sphere Sn.Lemma 2.1.1.1. If ? : Mm ? Sn ? Rn+1 is an isometric immersion, then?M? = sH = H ?m?,where H, sH are the mean curvature vectors of M in Sn, Rn+1 respectively.132.1. Minimal Submanifolds of SnHence, ? is a minimal immersion if and only if?M? = ?m?.Note 2.1.1.1. Here we think of ? as the vector ?i ??xi ? Rn+1 and by ?M?we mean the vector (?M?1,?M?2, . . . ,?M?n+1).Proof of 2.1.1.1: Let D,? and ? denote the covariant differentiation op-erators on Rn+1, Sn and M respectively, and let {Ei}mi=1 be a local orthonor-mal frame tangent to M . Then, if {y1, . . . , ym} and {x1, . . . , xn+1} are lo-cal coordinates on M and Rn+1 respectively, then we first note that, forf ? C?(Rn+1),d?(Ek)(f) = Ek(f ? ?) = Eik??yi(f ? ?) = Eik?f?xj??j?yi,where Ek = Eik??xi , and so we get that d?(Ek) = Ek (?j)??xj . From this wesee thatEk(?i)??xi= d?i(Ek)??xi= d?(Ek) = Ek,so Ek(?i) = Eik. HenceEkEk(?i)??xi= Ek(Eik) ??xi= dEik(Ek)??xi= dEk(Ek) = DEkEk.Also,(?EkEk) (?i)??xi= d?i (?EkEk)??xi= d? (?EkEk) = ?EkEk.Therefore,?M? =m?k=1EkEk(?)? (?EkEk)? =m?k=1DEkEk ??EkEk=m?k=1(DEkEk)? = sH,(2.1)where (?)? denotes the projection onto the space normal to TpM in TpRn+1.142.1. Minimal Submanifolds of SnNowH =m?k=1(?EkEk)N=m?k=1((DEkEk)T)N=m?k=1((DEkEk)N)T= ( sH)T ,where (?)N denotes the projection onto the space normal to TpM in TpSn,and (?)T denotes the tangential projection onto TpM . From this and (2.1)we see that?M? = sH = ( sH)T + ( sH)? = H + ??,for some function ?. Together ??? ? 1 and (1.3) give us that0 =12?M???2 = ??,?M??+ ?gradM??2 = ?+ ?gradM??2.Thus ? = ??gradM??2. Now?gradM??2 =n+1?i=1?gradM?i?2 =?i=1,...,n+1j=1,...,m?gradM?i, Ej?2=?i=1,...,n+1j=1,...,m?gradM?i, Ej?2=?i=1,...,n+1j=1,...,m(d?i(Ej))2 .If we choose the local coordinates {xi} so that the vectors { ??xi??p} are or-thonormal, then (at p) this gives us that?gradM??2 =?i=1,...,n+1j=1,...,m(d?i(Ej))2 =?i=1,...,n+1j=1,...,m?d?i(Ej)??xi, d?i(Ej)??xi?=m?j=1?d?(Ej)?2=m?j=11 = m,152.1. Minimal Submanifolds of Snand so we get that ?M? = sH = H ?m?.2.1.2 Simons Classification ResultThe easiest example of a closed minimal submanifold of Sn is the embeddedtotally geodesic Sk ? Sn, but these are by no means the only ones: in 1970,Lawson [8] showed that every compact, orientable surface can be minimallyembedded into S3. However, it is well known that the embedded, totallygeodesic Sk are the only immersed totally geodesic minimal submanifolds ofSn. We will briefly outline the proof of this fact before turning our attentionto the proof of the following main result: in 1968, Simons [16] proved thatany immersed minimal submanifold Mk of Sn is unstable (ind(M) ? 1), andthat ind(M) = 1 if and only if M is diffeomorphic to Sn?1 embedded in thestandard way as a totally geodesic submanifold. In fact, he proved that ifMk is a k-dimensional minimal submanifold of Sn, then ind(M) ? n?k withequality holding if and only if M is diffeomorphic to Sk (embedded into Sn inthe standard, totally geodesic way), classifying the totally geodesic spheresas those which minimize the index.Since we will be primarily concerned with characterizing the minimalsubmanifolds M of Sn which minimize the index, note that, henceforth,we will assume that M is connected. If M is not connected, then eachof its connected components is itself a minimal submanifold of Sn and theindex of M is the sum of the indices of its connected components. Since,as we will show, each connected minimal submanifold of Sn is unstable, itfollows that the index of M is strictly greater than the index of any of itsconnected components, and so characterizing the minimal submanifolds Snwhich minimize the index will be the same as characterizing the connectedminimal submanifolds of Sn which minimize the index.Lemma 2.1.2.1. Let Mk be a closed totally geodesic immersed submanifoldof Sn. Then M is isometric to Sk.Proof of 2.1.2.1: Let p ?M and v ? TpM . As M is assumed to be totallygeodesic, for sufficiently small t, expMp (tv) is a geodesic in Sn and thereforea piece of a great circle. Since M is closed, it is compact and therefore162.1. Minimal Submanifolds of Sncomplete, and ?M = ?. Hence expMp (vt) is defined for all t and must sweepout the entire great circle. So, letting v vary throughout TpM and t ? R,expMp (vt) sweeps out a k-dimensional sphere. Since M is connected, we musttherefore have that M ?= Sk.We now present the proof of the main classification result.Theorem 2.1.2.2. (Simons, 1968) Let M be a compact, closedk-dimensional minimal submanifold immersed in Sn. Then ind(M) ? n?k,with equality holding if and only if M is diffeomorphic to Sk embedded intoSn in the standard way as a totally geodesic submanifold.Proof of 2.1.2.2: For the sake of brevity, by Sk we will always mean thetotally geodesic minimal embedding Sk ?? Sn.We start by showing that ind(Sk) = n? k. First, for any k-dimensionalminimal submanifold we have that for any V ? ?(NSk),R(V ) := Tr[RSn(?, V )?] = ?kV. (2.2)To show this, we make note that, since Sn has constant sectional curvatureK ? 1, it follows thatRSn(X,Y, Y,X) = 1 ? (?X,X??Y, Y ? ? ?X,Y ?2)for all X,Y ? ?(T?M). Moreover,?X,W ??Y, Z? ? ?X,Y ??W,Z?is a covariant 4-tensor on TpSn with the same symmetries as the curvaturetensor. By a standard result in Riemannian geometry it must therefore beequal to the curvature tensor, i.e.RSn(X,Y, Z,W ) = ?X,W ??Y, Z? ? ?X,Y ??W,Z?for all X,Y, Z,W ? ?(TSn) (for proof see [10] (pg. 146) ). Hence it follows172.1. Minimal Submanifolds of SnthatRSn(X,Y )Z = ?Y,Z?X ? ?X,Z?Y,for all X,Y, Z ? ?(TSn). So, if {Ei}ki=1 is an orthonormal frame for TSk,thenR(V ) =k?i=1(RSn(Ei, V )Ei)N =k?i=1(?V,Ei?Ei ? ?Ei, Ei?V )N = ?kV.Since A ? 0 in the case of Sk (Sk is totally geodesic), this means that(1.6) becomesL = ?Sk + k,and so it is clear that the eigenspaces of ?Sk with eigenvalue ? correspondto eigenspaces of L with eigenvalue ??k. Now, note that the normal bundleof Sk is trivial: we can rotate, if necessary, so thatSk = {(x1, . . . xn+1) ? Sn ? Rn+1??xk+2 = . . . = xn+1 = 0}.Then the vectors { ??xk+2 , . . . ,??xn+1 } form a global frame for the normalbundle of Sk in Sn. From this it follows that we can choose a global parallelframe {V1, . . . , Vn?k} for NSk. Then, for any V ? ?(NSk), we can writeV = viVi for some function vi ? C?(Sk), and, since {Vi} is a parallel frame,?SkV = ?Sk(viVi) = ?Sk(vi)Vi.Hence, ?kS(V ) + ?V = 0 if and only if(?Sk(v1), . . . ,?Sk(vn?k))+ ?(v1, . . . , vn?k)= ~0. (2.3)Now, on functions, from Lemma 1.2.3.1 we know that ?Sk has a 1-dimensional nullspace and a (k + 1)-dimensional eigenspace correspondingto the eigenvalue k. Moreover, from Lemma 1.2.3.1 it is also clear that all ofthe other eigenvalues are all strictly greater than k. Thus, ind(Sk) = n? k.We will prove the second half of the theorem through a sequence of182.1. Minimal Submanifolds of Snlemmas which together establish that the index form Q takes a special formwhen restricted to a specific subspace of ?(NM). In particular, we will showthat for every u in this specific subspace of ?(NM),Q(u, u) = ?k?M?u?2 dV.It is clear from this that the index form is negative definite on this particularsubspace, and so the result will then follow once we show that the dimensionof this subspace is greater than or equal to n? k, with equality if and onlyif A ? 0.As before, let D,?,? denote the Levi-Civita connections for Rn+1, Snand M respectively, and ?N ,?Ndenote the connections on the normalbundles of M in Sn and of Sn in Rn+1 respectively.Given a parallel vector field in Rn+1, we can take the tangential projec-tion onto Sn to get a vector field on Sn. The collection of all such vectorfields, ?, forms an (n+ 1)-dimensional subspace of ?(TSn).Lemma 2.1.2.3. Let Z ? ?, p ? Sn. Then there is ? ? C?(Sn) such that,for any X ? TpSn?XZ = ?X.Proof of 2.1.2.3: Let Z = W T , where W is a parallel vector field on Rn+1so that?XZ = (DXZ)T =(DXWT )T =(DX(W ?WN))T= ?(DXWN)T ,(2.4)where WN is the component of W normal to Sn. If we let N be the unitnormal to Sn, then WN = ??N for some function ?, and so putting thisinto (2.4) we get that?XZ = (DX?N)T = (X(?)N + ?DXN)T = ?X,where, as a consequence of the standard coordinate vector fields on Rn+1192.1. Minimal Submanifolds of Sn{?i := ??xi }n+1i=1 being parallel, we have thatDXN = XiD?i(xj?j)= Xi?i = X.Now, since Z ? ? is a vector field on Sn, restricting it to M and takingthe normal and tangential projections gives vector fields ZN ? ?(NM) andZT ? ?(TM) respectively.Lemma 2.1.2.4. For Z ? ?, the vector fields ZN ? ?(NM) and ZT ??(TM) satisfy?NXZN = ?A(X,ZT ),?XZT = ?(?XZN)T + ?X,where X ? TpM and ? is independent of X.Proof of 2.1.2.4: Taking into account Lemma 2.1.2.3 we have that?NXZN =(?XZN)N =(?X(Z ? ZT))N=(?X ??XZT )N= ?A(X,ZT ).Again, using Lemma 2.1.2.3 we see that?XZT =(?XZT )T =(?X(Z ? ZN))T= ?X ?(?XZN)T .Let {Ei}ki=1 be a local orthonormal frame for TM for which ?EiEj |p = 0for i, j = 1, . . . , k. Though we will not write it explicitly, note that thefollowing calculations will all be computed at p.Lemma 2.1.2.5. Let Z ? ? and consider ZN ? ?(NM). Then??NZN?2 =k?i=1?ZN , A(Ei,?EiZT )?+ Ei?ZN ,?NEiZN ?.202.1. Minimal Submanifolds of SnProof of 2.1.2.5: Using Lemma 2.1.2.4 we have that??NZN?2 =k?i=1??NEiZN ,?NEiZN ?=k?i=1Ei?ZN ,?NEiZN ? ? ?ZN ,?Ei?NEiZN ?=k?i=1Ei?ZN ,?NEiZN ?+ ?ZN ,?NEi(A(Ei, ZT ))?Now?NEi(A(Ei, ZT ))=(?EiA)(Ei, ZT ) +A(?EiEi, ZT ) +A(Ei,?EiZT )=(?EiA)(Ei, ZT ) +A(Ei,?EiZT ).(2.5)From the Codazzi Equation (1.1.2.2),(?EiA)(ZT , Ei) =(?ZTA)(Ei, Ei) +(RSn(Ei, ZT )Ei)N,and, again using that Sn has constant curvature,(RSn(Ei, ZT )Ei)N=(?ZT , Ei?Ei ? ?Ei, Ei?ZT )N = 0.Also,(?ZTA)(Ei, Ei) = ?NZT (A(Ei, Ei))?A(Ei,?ZTEi)?A(?ZTEi, Ei)= ?NZT (A(Ei, Ei)) ,since ?ZTEi =?kj=1?ZT , Ej? ? ?EjEi = 0. Thusk?i=1(?ZTA)(Ei, Ei) =k?i=1?NZT (A(Ei, Ei))= ?NZTk?i=1A(Ei, Ei) = ?NZTH = 0,212.1. Minimal Submanifolds of Snand so we get that(?EiA)(ZT , Ei) = 0. Therefore (2.5) becomes?NEi(A(Ei, ZT ))= A(Ei,?EiZT ).Hence,??NZN?2 =k?i=1Ei?ZN ,?NEiZN ?+ ?ZN ,?NEi(A(Ei, ZT ))?=k?i=1?ZN , A(Ei,?EiZT )?+ Ei?ZN ,?NEiZN ?Lemma 2.1.2.6. Let Z ? ? and consider ZN ? ?(NM). Then??NMZN , ZN ? = ??A?(ZN), ZN ?.Proof of 2.1.2.6: First we have that??NZN?2 =k?i=1?(?EiZN)N ,(?EiZN)N ?=k?i=1Ei??EiZN , ZN ? ? ?ZN ,?Ei?NEiZN ?= ??ZN ,?NMZN ?+k?i=1Ei??EiZN , ZN ?.So, Lemma 2.1.2.5 tells us that??NMZN , ZN ? =k?i=1??ZN , A(Ei,?EiZT )?. (2.6)222.1. Minimal Submanifolds of SnNow, using Lemma 2.1.2.4 we also have that?EiZT =k?j=1?Ej ,?EiZT ?Ej =k?j=1??Ej ,(?EiZN)T ?Ej + ??i,jEj=k?j=1??EiEj , ZN ?Ej + ??i,jEj=k?j=1?A(Ei, Ej), ZN ?Ej + ??i,jEj .So, since A is bilinear,A(Ei,?EiZT ) =k?j=1?A(Ei, Ej), ZN ?A(Ei, Ej) + ??i,jA(Ei, Ej),and thereforek?i=1?ZN , A(Ei,?EiZT )? =k?i,j=1?A(Ei, Ej), ZN ?2 + ??i,j?A(Ei, Ej), ZN ?= ?A?(ZN ), ZN ?+k?i=1??A(Ei, Ei), ZN ?= ?A?(ZN ), ZN ?+ ??H,ZN ?= ?A?(ZN ), ZN ?.Hence, (2.6) becomes??NMZN , ZN ? = ??A?(ZN ), ZN ?.Lemma 2.1.2.7. For Z ? ?, considering ZN ? ?(NM) we have thatQ(ZN , ZN ) = ?k?M??ZN??2 dV.232.1. Minimal Submanifolds of SnProof of 2.1.2.7: Since (2.2) holds for any k-dimensional minimal sub-manifold of Sn, (1.6) becomesLZN = ?M(ZN)+ kZN + A?(ZN ).Hence, the desired result follows from applying Lemma 2.1.2.6.Let ?N = {ZN ? ?(NM) | Z ? ?}. Then it follows from Lemma 2.1.2.7that Q(?, ?) is negative definite on ?N .Lemma 2.1.2.8. We have that dim ?N ? n? k with equality if and only ifM is isometric to Sk (embedded in the standard way as a totally geodesicsubmanifold).Proof of 2.1.2.8: It is clear that at each p ? M , ?(p) := {X(p) | X ? ?}spans TpSn, and so for each p ? M , ?N spans NpM ? TpSn. Hence,dim ?N ? dimNpM = n? k.Now suppose that dim ?N = n ? k, and let ? be the kernel of homo-morphism ? ? ?N so that if X ? ?, then XT = X at every p ? M , anddefine ?(p) := {X(p) | X ? ?} . Now if we let ?p be the kernel of the homo-morphism ? ? NpM defined by Z 7? ZN (p) (p ? M), then it is clear that?(p) ? ?p. So since dim ?p = (n + 1) ? (n ? k), and since our assumptiondim ?N = n ? k implies that dim ? = (n + 1) ? (n ? k), we must have that? = ?p. This means that the map Z 7? ZT (p), which clearly maps ?p ontoTpM , must map ? onto TpM . Hence, for any Z ? TpM , there exists Z? ? ?such that Z?(p) = Z, and Z? is everywhere tangent to M . From this andLemma 2.1.2.4 it follows that, for any X,Z ? TpM ,A(X,Z) = ??NX Z?N = 0,and so we can conclude that A ? 0. Thus M is totally geodesic and thereforeisometric to Sk by Lemma 2.1.2.1.242.2. Non-totally Geodesic Minimal Hypersurfaces of Sn2.2 Non-totally Geodesic Minimal Hypersurfacesof SnIf we restrict our attention to non-totally geodesic minimal hypersurfacesof Sn, then it is conjectured that the index also characterizes the Cliffordhypersurfaces in a way similar to Simon?s classification result for minimalhypersurfaces of index one: the Clifford hypersurfaces are thought to mini-mize the index amongst non-totally geodesic minimal hypersurfaces. Indeedwe will prove that this is the case when n = 3.2.2.1 A Classification ConjectureHere we introduce the Clifford hypersurfaces of Sn and some of their prop-erties before turning to the classification conjecture.Definition 2.2.1.1. By a Clifford hypersurface of Sn we mean a productSk(?kn?1)? Sl(?ln?1), where k + l = n? 1.To show that the Clifford hypersurfaces are, in fact, minimal, we followLawson?s exposition [9].Lemma 2.2.1.1. The Clifford hypersurfaces are minimal submanifolds ofSn.The proof can be found in [9] (pg. 23).The next theorem gives a characterization of the Clifford hypersurfacesin terms of the square norm of their second fundamental form. We will usethis result in calculating their index, as well as in the proof of the conjecturein the case n = 3 (Theorem 2.2.2.5). The proof can be found in [5] (see theMain theorem, pg. 60).Theorem 2.2.1.2. (Chern, Do Carmo, Kobayashi (1970); Lawson (1969))The Clifford hypersurfaces are the only minimal hypersurfaces of Sn with?A?2 ? n? 1.As mentioned above, it is thought that the Clifford hypersurfaces arealso characterized by their index.252.2. Non-totally Geodesic Minimal Hypersurfaces of SnConjecture 2.2.1.1. Let M be a closed, orientable, non-totally geodesicminimal hypersurface of Sn. Then ind(M) ? n + 2 with equality holding ifand only if M is a Clifford hypersurface.We now show that one direction of the conjecture is true: the Cliffordhypersurfaces have index n+ 2.Lemma 2.2.1.3. If M = Sk(?kn?1)? Sl(?ln?1)is a Clifford hypersur-face where k + l = n? 1, then ind(M) = n+ 2.Proof of 2.2.1.3: It follows from Theorem 2.2.1.2 that L = ?M + 2(n ?1), and so the eigenvalues of L are in one-to-one correspondence with theeigenvalues of ?M : ?Mf + ?f = 0 if and only if Lf + ?f = 0 where ? =??2(n?1). Now, since M is a product manifold (with the product metric),we can use Lemma 1.2.3.2 to calculate the eigenvalues of the Laplacian on Mif we know the eigenvalues of the Laplacian on Sk(?kn?1)and Sl(?ln?1).From Lemma 1.2.3.1 we know that the eigenvalues of the Laplacian onSk(1) are?j = (j ? 1)(k + j ? 2), j = 1, 2, 3, . . .= 0, k, 2(k + 1), . . .(2.7)with multiplicitydimPj?1 ? dimPj?3 =(k + j ? 1j ? 1)?(k + j ? 3j ? 3)= 1, k + 1,(k + 22)? 1, . . .(2.8)where Pj is the space of homogeneous polynomials of degree j on Rk+1.Now let f :(Sk(1), h)?(Sk(r), g)be defined by x 7? rx, where g is themetric induced from Rk+1 and h is the pullback metric h = f?g. Clearlyh = r2g is a scaling of the metric on Sk(1) induced from Rk+1, so we can findthe eigenvalues of the Laplacian on Sk(r) by figuring out how the eigenvaluesof the Laplacian on Sk(1) change when we scale the metric. Using the local262.2. Non-totally Geodesic Minimal Hypersurfaces of Snformula for the Laplacian with respect to the metric h, ?h, we see that?hf =1?H??xi(hij?H??xjf)=1rk?G??xi(r?2gijrk?G??xjf)=1r2?gf,where?H =?|det(hij)|. This shows that?gf + ?f = 0 ?? ?hf +?r2f = 0,and so scaling the metric by a factor of r2 has the effect of scaling the eigen-values by a factor of 1r2 . Hence, from (2.7) and (2.8) the first three eigenval-ues of the Laplacian on Sk(?kn?1)and Sl(?ln?1)are ?k,1 = 0 (multiplic-ity 1), ?k,2 =k(n?1)k = n?1 (multiplicity k+1), ?k,3 =2(k+1)(n?1)k (multiplic-ity(k+22)? 1) and ?l,1 = 0 (multiplicity 1), ?l,2 =l(n?1)l = n? 1 (multiplic-ity l + 1), ?l,3 =2(l+1)(n?1)l (multiplicity(l+22)? 1) respectively. Therefore,Lemma 1.2.3.2 implies that the first three eigenvalues of the Laplacian on Mare ?1 = 0 (multiplicity one), ?2 = n?1 (multiplicity (k+1)+(l+1) = n+1)and ?3 = min{2(n? 1), 2(k+1)(n?1)k ,2(l+1)(n?1)l}= 2(n ? 1). From this itfollows that the first two eigenvalues of L on M are ?1 = ?2(n ? 1) (mul-tiplicity 1) and ?2 = ?(n ? 1) (multiplicity n + 1), and that all the othereigenvalues are non-negative. Hence ind(M) = n+ 2.The other direction of the conjecture is much less clear. Though it isknown to be true in the case n = 3 (see ?2.2.2), there are several thingsused in the proof of this special case which are no longer available whenone moves to higher dimensions, preventing the proof from generalizing.Perdomo [14] has shown that the conjecture is true under an additionalsymmetry assumption which all known minimal hypersurfaces of spheressatisfy. Namely, he has proved the following theorem:Theorem 2.2.1.4. (Perdomo, 2001) Let M be a compact, oriented non-totally geodesic minimal hypersurface of Sn, and let OM (n+ 1) be the sub-group of the orthogonal group O(n+ 1) consisting of orthogonal transforma-272.2. Non-totally Geodesic Minimal Hypersurfaces of Sntions which fix M , i.e.OM (n+ 1) = {? ? O(n+ 1) | ?(M) = M}.If OM (n+1) fixes only the origin of Rn+1, then ind(M) ? n+2 with equalityif and only if M is a Clifford hypersurface.His proof heavily relies on the imposed symmetry condition, and it seemsthat the only obvious way to extend his result to prove the full conjecturewould be to somehow show that all minimal submanifolds of Sn of indexn+ 2 actually satisfy the symmetry condition.Motivated by the behaviour of the index functional of the space of min-imal hypersurfaces with constant squared norm of the second fundamentalform, Perdomo [14] has also conjectured that ind(M) ? 2n+ 3 for all non-totally geodesic minimal hypersurfaces of Sn which are not Clifford. This istrue for all known examples of minimal hypersurfaces of Sn, and Perdomoshows that it is also true when one also assumes that the symmetry condi-tion in Theorem 2.2.1.4 is satisfied and that the hypersurfaces has an evenunit normal vector N in the sense that the functions fa(p) = ?a,N(p)? areall even functions.2.2.2 Urbano?s Result: Proof of the Conjecture when n = 3In 1990, Urbano proved that the conjecture is true in the special case ofn = 3. We now mention those results that will be used in the proof thathave not already been stated.Lemma 2.2.2.1. (Almgren, 1966) Let f : S2 ? S3 ? R4 be a real analyticminimal immersion. Then f embeds S2 in S3, and f(S2) = S3 ? {x ?S3 | x ? v = 0} for some v ? S3.The proof can be found in [1] (pg. 279).Lemma 2.2.2.2. (Obata, 1962) A complete Riemannian manifold Mn withn ? 2 admits a non-constant function ? such that Hess?(?, ?) = ?c2???, ?? ifand only if M is isometric with a sphere Sn(1c)in Rn+1.282.2. Non-totally Geodesic Minimal Hypersurfaces of SnThe proof can be found in [13] (pg. 334).Theorem 2.2.2.3. (Gauss-Bonnet) Suppose M is a compacttwo-dimensional Riemannian manifold. Then?MK dA = 2pi ? ?(M),where ?(M) is the Euler characteristic of M .Lemma 2.2.2.4. Let M be a 2-dimensional Riemannian manifold with localconformal parameter z = x + iy and let f : M ? ?M be a conformal map.ThenArea (f(M)) =12E(f),where Area (f(M)) is the area of f(M) ? ?M and E(f) is the energy of f ,E(f) =?M?df, df?.Proof of 2.2.2.4: Let g, also written as ??, ??, be the metric on ?M so thatf?g is the metric on M . Let {E1, E2} be a local orthonormal frame for TM ,and let {?1, ?2} be the coframe for T ?M dual to {E1, E2}. Then, since f isconformal and therefore preserves angles, the energy of f is simplyE(f) =?M(?df(E1), df(E1)?+ ?df(E2), df(E2)?) d?1 ? d?2. (2.9)Moreover, since E1 and E2 each have unit length and f is conformal,?df(E1), df(E1)? = ?df(E2), df(E2)? =: ?.Hence, (2.9) simplifies toE(f) = 2?M? d?1 ? d?2. (2.10)292.2. Non-totally Geodesic Minimal Hypersurfaces of SnOn the other hand the area is given byArea(f(M)) =?MdV =?M?det(f?g) d?1 ? d?2,where (f?g)ij = ?df(Ei), df(Ej)?. Then since det(f?g) = ?2, using (2.10) itis easy to see thatArea(f(M)) =?M? d?1 ? d?2 =12E(f).Amongst the various tools that Urbano uses, some are are only availablein the specific case when M is a surface; the Gauss-Bonnet Formula and theresult of Almgren are only available in dimension two and don?t allow theproof to be directly generalized to higher dimensions.Theorem 2.2.2.5. (Urbano, 1990) Let M be a compact orientable non-totally geodesic minimal surface is S3. Then ind(M) ? 5, and the equalityholds if and only if M is the Clifford torus.The proof is roughly broken into three parts. First we argue thatind(M) ? 5 by showing that ?2 is an eigenvalue of L and construct aneigenspace corresponding to the eigenvalue ?2. Using the fact that M isassumed to be non-totally geodesic, we show that this eigenspace has di-mension 4. Since the first eigenvalue of L is simple, this gives us the desiredresult. Next we assume that ind(M) = 5. Our previous calculations stillhold and therefore tell us that the second eigenvalue is ?2 and that all othersare non-negative. We use the Hersch trick to construct a conformal transfor-mation Fg of M for which the component functions of Fg ?? are orthogonalto the first eigenfunction, where ? : M ? S3 is a minimal immersion. Wethen show that equality actually holds in (1.13), and using Lemma 2.1.1.1we can then show that ?A?2 ? 2. It then follows from Theorem 2.2.1.2 thatM must be the Clifford torus. The final step in the proof is to show thatthe Clifford torus actually has index 5, but this follows from Lemma 2.2.1.3.As with Simons?s classification result, the details of the proof will be302.2. Non-totally Geodesic Minimal Hypersurfaces of Snestablished through a sequence of lemmas.Proof of 2.2.2.5: Let ? : M ? S3 be a minimal immersion, N denote the unitnormal vector field to M , and let ?,? and D denote the covariant differenti-ation operators on M,S3 and R4 respectively (?XY = (?XY )TM , ?XY =(DXY )TS3 ).Lemma 2.2.2.6. For any vector a ? R4, the function fa = ?a,N? is aneigenfunction of L with eigenvalue ?2.Proof of 2.2.2.6: Let p ? M and {E1, E2} be an orthonormal frame in TM(defined in a neighbourhood of p) such that ?EiEj(p) = 0 for i, j = 1, 2.Then, {E1, E2, N,?} is a local orthonormal frame for R4, and?Mfa =2?i=1EiEi(fa)??EiEifa =2?i=1EiEi?a,N?(note that this calculation is evaluated at p and that we will continue tosuppress the p for convenience). Since a ? R4, ??, ?? here denotes the metricon R4, and so by the compatibility of the R4 metric with D we have that?Mfa =2?i=1Ei (?DEiN, a?+ ?N,DEia?) =2?i=1Ei?DEiN, a?,andDEiN = ?DEiN,E1?E1 + ?DEiN,E2?E2 + ?DEiN,N?N + ?DEiN,???.Since N is a unit vector, ?DEiN,N? =12Ei?N,N? = 0. Also, if X and Y areorthogonal vector fields on R4, then from the compatibility of the connectionwith the inner product we have that for any vector field Z on R4.?DZX,Y ? = Z?X,Y ? ? ?X,DZY ? = ??X,DZY ?. (2.11)312.2. Non-totally Geodesic Minimal Hypersurfaces of SnNow ?DEiN,?? = ??N,DEi?? = ??N,Ei? = 0, soDEiN =2?j=1?DEiN,Ej?Ej = ?2?j=1?N,DEiEj?Ej (by (2.11))= ?2?j=1hijEj ,(2.12)where hij = ?N,?EiEj? = ?N,DEiEj? is the scalar second fundamentalform of M in S3. Thus,?Mfa = ?2?i,j=1Ei (hij?Ej , a?)=2?i,j=1Ei(hij)?Ej , a?+ hij?DEiEj , a?+ 0.(2.13)From the Codazzi Equation (Lemma 1.1.2.2) we know that, sinceRR4 = 0,0 = ?RR4(Ei, Ej)Ek, N? = ?(DEj sA)(Ei, Ek), N? ? ?(DEi sA)(Ej , Ek), N?,(2.14)where sA is the second fundamental form on M as a submanifold of R4 and?(DEi sA)(Ej , Ek), N? = Ei? sA(Ej , Ek), N? ? ? sA(?EiEj , Ek), N?? ? sA(Ej ,?EiEk), N? ? ? sA(Ej , Ek), sA(Ei, N)?.From (2.12) we see that DEiN is tangent to M , so? sA(Ej , Ek), sA(Ei, N)? = ?(DEjEk)?, (DEiN)?? = ?(DEjEk)?, 0? = 0.Also, since ?EiEj = 0 (at p) for all i, j, andEi(hjk) = Ei?DEjEk, N? = Ei?(DEjEk)?, N? = ?(DEi sA)(Ej , Ek), N?,322.2. Non-totally Geodesic Minimal Hypersurfaces of Sn(2.14) yieldsEi(hjk) = ?(DEi sA)(Ej , Ek), N? = ?(DEj sA)(Ei, Ek), N? = Ej(hik).Hence, the Codazzi Equation and the symmetric property of the secondfundamental form imply thatEi(hij) = Ei(hji) = Ej(hii). (2.15)Therefore2?i,j=1Ei(hij)?Ej , a? =2?i,j=1Ej(hii)?Ej , a? =2?j=1Ej(h11 + h22)?Ej , a? = 0,since M is minimal and therefore has zero mean curvature by Lemma 1.1.3.3.Thus, from (2.13) we find that?Mfa = ?2?i,jhij?DEiEj , a?. (2.16)Now,DEiEj = ?DEiEj , E1?E1 + ?DEiEj , E2?E2 + ?DEiEj , N?N + ?DEiEj ,???= ?DEiEj , N?N + ?DEiEj ,???,(2.17)since ?DEiEj , Ek? = ??EiEj , Ek? = 0. Also, using (2.11) we see that?DEiEj ,?? = ??Ej , DEi?? = ??Ej , Ei? = ??ji, (2.18)332.2. Non-totally Geodesic Minimal Hypersurfaces of Snand so (2.16) becomes?Mfa = ?2?i,jhij (hij?N, a? ? ?ji?)= ??A?2fa +(2?i=1hii)??, a?= ??A?2fa,again using the minimality of M and Lemma 1.1.3.3. Hence,Lfa? 2fa = (?Mfa + ?A?2fa + 2fa)? 2fa = (??A?2 + ?A?2 + 2? 2)fa = 0.Lemma 2.2.2.6 and the bilinearity of the metric imply that V = {fa | a ?R4} forms a subspace of the eigenspace of L associated to the eigenvalue ?2,and clearly dimV ? 4.If dimV ? 3, then the kernel of the linear transformation R4 ? V ,a 7? fa is non-empty, and so there is a non-zero a ? R4 for which fa ? 0.Lemma 2.2.2.7. If a ? R4, a 6= ~0, is such that fa ? 0, then g = ??, a?satisfies HessMg(?, ?) = ?g??, ??.Proof of 2.2.2.7:Let {E1, E2} be as in the proof of Lemma 2.2.2.6. ThenHessMg(Ei, Ej) = EiEj(g)? (?EiEj) (g)= EiEj??, a?= Ei (?DEi?, a?+ 0)= Ei?Ei, a?= ?DEjEi, a?.From (2.17) and (2.18) we know thatDEiEj = hijN ? ?ij?,342.2. Non-totally Geodesic Minimal Hypersurfaces of Snand soHessMg(Ei, Ej) = hij?N, a? ? ?ij??, a? = ??ijgsince fa = ?N, a? = 0 by choice of a. The desired result follows from thebilinearity of HessM and ??, ??.Now, from Obata?s Lemma 2.2.2.2 we have that either M is isometricto a unit sphere or g ? 0. Since Almgren showed that the all minimalspheres in S3 are totally geodesic (Lemma 2.2.2.1), it must be that g ? 0.Now, rotating M (if necessary) through an isometry r : S3 ? S3 so that acoincides with the standard basis vector e4 ? R4 we see that0 = g = ???, a? = ???, e4? = ??4,where ?? = r ? ? and ??4 = (r ? ?)4 is the fourth component function of ??.Hence,1 = ??21 + ??22 + ??23 + ??24 = ??21 + ??22 + ??23,and so we see that ??(M), and therefore ?(M), lies in an equator of S3 andis therefore totally geodesic. Hence, dimV 3 and therefore dimV = 4.Since the first eigenvalue of L is simple, we conclude that it cannot be?2 and that ind(M) ? 5.Now, if ind(M) = 5, then the second eigenvalue of L is ?2. Let ? > 0be an eigenfunction for the first eigenvalue ?1.Lemma 2.2.2.8. There is a conformal transformation Fg of S3 such that?M? ? (Fg ? ?)i dA = 0, i = 1, 2, 3, 4.Proof of 2.2.2.8: To prove the lemma we follow the exposition of Montieland Ros found in [11] (pg. 154). Consider the conformal transformations ofS3 of the formFg(p) =p+ (??p, g?+ ?)g?(1 + ?p, g?), (2.19)where ? = (1? ?g?2)?12 , ? = (?? 1)?g??2 and g ? B4 is fixed.352.2. Non-totally Geodesic Minimal Hypersurfaces of SnThe collection of these conformal transformations can be extended to?B4 = {x ? R4 | ?x? ? 1} by letting g ? ?B4. Note that if p is such that?p, g? 6= ?1, thenFg(p) =p+ ?(?g??2?p, g?+ 1)g ? ?p, g??g??2g?(1 + ?p, g?)= g +?1? ?g?2(p? ?p, g?g1 + ?p, g?)= g.(2.20)Consider the map H : ?B4 ? ?B4 whose component functions are givenbyHi(g) =1?M ? dA?M? ? (Fg ? ?)i dA,for i = 1, 2, 3, 4. We have that?H(g)? =1???M ? dA???????M? ? (Fg ? ?) dA?????1?M ? dA?M|?| ? ?Fg ? ?? dA=1?M ? dA?M? dA = 1 (since ? ? 0, ?Fg ? ?? = 1),and if g ? S3, (2.20) shows that Fg = g except on a set of measure zero, soH(g) =1?M ? dA?M? ? g dA = g.Thus H maps ?B4 to itself and is the identity on the boundary ??B4 = S3, soby a standard topological argument H must be surjective. Hence there is ag ? B4 such that H(g) = 0 and therefore?M? ? (Fg ? ?) dA = 0.362.2. Non-totally Geodesic Minimal Hypersurfaces of SnTogether Lemma 2.2.2.8 and (1.13) imply thatQ((Fg ? ?)i, (Fg ? ?)i) ? ?2 ??M(Fg ? ?)2i dA (2.21)for i = 1, 2, 3, 4, with equality holding if and only if (Fg ? ?)i is an eigen-function of L associated to the eigenvalue ?2.Note that (Fg ? ?)i 6? 0; if for some i, (Fg ? ?)i ? 0, then as arguedbefore Fg ??(M) lies in an equator of S3. Hence there is a non-zero vectorv ? R4 such that Fg ? ?(M) lies is the hyperspace orthogonal to v, and soM lies in the hyperspace orthogonal to F?1g (v) (since Fg preserves angles)and therefore lies in an equator of S3 and is totally geodesic. This meansthat if we get equality in (2.21), then we can divide both sides of the equa-tion by??(Fg ? ?)i??2L2 6= 0 and apply (1.13) to get that the L2-normalizedcomponent functions, and hence the unnormalized component functions, areeigenfunctions with eigenvalue ?2.Hence,?M??(Fg ? ?)i?2 ? (?A?2 + 2)(Fg ? ?)2i dA ? ?2?M(Fg ? ?)2i dAand so ?M??(Fg ? ?)i?2 dA ??M?A?2(Fg ? ?)2i dA,which gives?M??(Fg ? ?)?2 dA =4?i=1?M??(Fg ? ?)i?2 dA?4?i=1?M?A?2(Fg ? ?)2i dA =?M?A?2 dA,since?4i=1(Fg ? ?)2i ? 1.Lemma 2.2.2.9.2A(f(M)) =?M??(Fg ? ?)?2 dA+ 2?M(?N, g?1 + ??, g?)2dA,372.2. Non-totally Geodesic Minimal Hypersurfaces of Snwhere A(f(M)) is the area of M .Proof of 2.2.2.9: For g ? ?B4, let Fg denote the conformal transformationof S3 as in (2.19). Then, thinking of Fg as a map R4 ? R4, we have thatthe jth component of dFg(??xi)is(dFg(??xi))j=1?[(?ij + ?gigj)(?p, g?+ 1)? (pj + (??p, g?+ ?)gj)gi](?p, g?+ 1)2= (?p, g?+ 1)?2??1 [?ij?p, g?+ ?ij + gigj(?? ?)? pjgi]= (?p, g?+ 1)?2??2[??ij?p, g?+ ??ij + gigj(1? ?)?g??2??pjgi] ,sodFg(v) = vidFg(??xi)=4?i,j=1vi(?p, g?+ 1)?2??2[??ij?p, g?+ ??ij + gigj(1? ?)|g|?2 ? ?pjgi] ??xj= (?p, g?+ 1)?2??2[?v?p, g?+ ?v + ?v, g?g(1? ?)?g??2 ? ?p?v, g?].From this we get that, for v, w ? TpS3,?dFg(v), dFg(w)?= (?p, g?+ 1)?4??4[?(?p, g?+ 1){?(?p, g?+ 1)?v, w?+ ?w, g?(1? ?)?g??2?v, g? ? ??w, g??p, v?}+ ?v, g?(1? ?)?g??2{?(?p, g?+ 1)?g, w?+ ?w, g?(1? ?)?g??2?g, g???w, g??p, g?}? ??v, g?{?(?p, g?+ 1)?p, w?+ ?w, g?(1? ?)?g??2?p, g? ? ??w, g??p, p?}]382.2. Non-totally Geodesic Minimal Hypersurfaces of Sn=(?p, g?+ 1)?4??4[?v, w??2(?p, g?+ 1)2+ ?v, g??w, g?{?(?p, g?+ 1)(1? ?)?g??2+ (1? ?)?g??2 ? ?(1? ?)?g??2?p, g?+ ?2?p?2}]=1? ?g?2(?p, g?+ 1)2 ?v, w?.So, for an isometric immersion ? : M ? S3, the area of Fg ? ?, Area(g), isgiven byArea(g) =?M1? ?g?2(??, g?+ 1)2 dA. (2.22)Now, for fixed g ? ?B4, define f : M ? R by f = ??, g?+ 1. Then, if welet {E1, E2} be a local orthonormal frame for TM such that ?EiEj??p = 0,we have that (at p)?M log f =2?i=1EiEi(log f) =2?i=1Ei[1fEi(f)]= ?1f2[E1(f)2 + E2(f)2]+1f?M (f)=1f2[?E1(f)2 ? E2(f)2 + f?M (f)].(2.23)Also,Ei(f) = Ei(??, g?+ 1) = ?DEi?, g?+ 0 + 0 = ?Ei, g?. (2.24)Now g = gN + gT , where gN is the component of g normal to M (andtangent to S3) and gT is the component tangent to M , and gT = ?g,E1?E1+?g,E2?E2. Hence?gT ?2 = ?g,E1?2 + ?g,E2?2. (2.25)392.2. Non-totally Geodesic Minimal Hypersurfaces of SnAlso,?Mf =2?i=1EiEi??, g? =2?i=1?DEiDEi?, g? = ??M?, g?. (2.26)Taking this into account and using Lemma 2.1.1.1 we see thatf?M (f) = (??, g?+ 1) ??M?, g?= (??, g?+ 1) ??2? +H, g?= ?2??, g?2 ? 2??, g?+ f?H, g?,(2.27)where H is the mean curvature vector of M in S3. So (2.23), (2.24), (2.25)and (2.27) give?M log f = f?2[?2??, g?2 ? 2??, g?+ f?H, g? ? ?gT ?2]. (2.28)Since ?gT ?2 = ?g?2 ? ?gN?2 ? ?g, ??2, (2.28) simplifies to?M log f = f?2 [???, g?2 ? 2??, g?+ f?H, g? ? ?g?2 + ?gN?2]= f?2[?(1 + ??, g?)2 + f?H, g?+ ?gN?2 + (1? ?g?2)]= ?1 +1? ?g?2f2+f?H, g?+ ?gN?2f2(2.29)Therefore, using (1.4) and (2.22) we have that0 =?M?M log f dA =?M?1 +1? ?g?2f2+f?H, g?+ ?gN?2f2dA= ?Area(f(M)) + Area(g) +?Mf?H, g?+ ?gN?2f2dA.If ? is, moreover, a minimal immersion, then H ? 0 and the equationsimplifies toArea(f(M)) = Area(g) +?M?gN?2f2dA. (2.30)So, for our minimal immersion ? : M ? S3, (2.30) holds. Now, since Fg402.2. Non-totally Geodesic Minimal Hypersurfaces of Snis conformal, from Lemma 2.2.2.4 we get that the area, Area(g), is equal tohalf of the energy of Fg ? ? and so2Area(f(M)) =?M?? (Fg ? ?) ?2 dA+ 2?M?gN?2f2dA.From Lemma 2.2.2.9 it is easy to see that2Area(f(M)) ??M??(Fg ? ?)?2 dA ??M?A?2 dA, (2.31)with equality if and only if ?N, g? ? 0 and we have equality in (2.21).Lemma 2.2.2.10. If K is the Gauss curvature of M , then?A?2 = 2? 2K.Proof of 2.2.2.10: Let {E1, E2} be as before. Then?A?2 =2?i,j=1?A(Ei, Ej), A(Ei, Ej)?=2?i,j=1?(?EiEj)N, N?2= h211 + h212 + h221 + h222,where hij =??EiEj , N?. Since M is minimal and A is symmetric thissimplifies to?A?2 = 2(h212 ? h11h22). (2.32)Moreover, using the Gauss equation (1.1.2.1) we see thath11h22 =?A(E1, E1), N??A(E2, E2), N?=?A(E1, E1),?A(E2, E2), N?N?=?A(E1, E1), A(E2, E2)?=?sR(E1, E2)E1, E2???R(E1, E2)E1, E2?+?A(E1, E2), A(E1, E2)?=?sR(E1, E2)E1, E2???R(E1, E2)E1, E2?+ h212.412.2. Non-totally Geodesic Minimal Hypersurfaces of SnHence, using this with (2.32) we get2(h212 ? h11h22) = 2( ?sR(E1, E2)E2, E1??E1?2?E2?2 ? ?E1, E2?2??R(E1, E2)E2, E1??E1?2?E2?2 ? ?E1, E2?2)= 2? 2K.From the Gauss-Bonnet formula (Lemma 2.2.2.3) we get?MK dA = 2pi ? ?(M) = 2pi(2? 2?),where ? is the genus of M . Therefore, (2.31) and Lemma 2.2.2.10 togethergive2Area(f(M)) ??M?A?2 dA =?M2? 2K dA= 2Area(f(M))? 2?MK dA= 2Area(f(M))? 4pi(2? 2?),and so we see that ? ? 1. Again, Almgren?s result (Lemma 2.2.2.1) showsthat if ? = 0, then M is totally geodesic, so it must be that ? = 1 andequality holds in the above inequalities. That is, ?N, g? ? 0 and equalityholds in (2.21).If we let f = ??, g?, then it follows from Lemma 2.2.2.7 that Hess f(?, ?) =?f??, ??. Using the same argument as before, we can conclude that g = 0and so Fg ? ? = ?. Therefore, having equality in (2.21) implies that thecomponent functions ?i, i = 1, 2, 3, 4, are eigenfunctions of L with eigenvalue?2. Hence,L?i ? 2?i = 0 for i = 1, 2, 3, 4,so?M?i = ??A?2?i for i = 1, 2, 3, 4.However, from Lemma 2.1.1.1 we also know that ? : M2 ? S3 is a minimal422.2. Non-totally Geodesic Minimal Hypersurfaces of Snimmersion if and only if ?M? = ?2? (?M?i = ?2?i for i = 1, 2, 3, 4).Therefore?i(?A?2 ? 2) = 0 for i = 1, 2, 3, 4,and since for any p ? M there is an i such that ?i(p) 6= 0, we must havethat ?A?2 ? 2. By Theorem 2.2.1.2 this means that M must be the Cliffordtorus.It follows from Lemma 2.2.1.3 that the index of the Clifford torus is 5.43Chapter 3ConclusionIn this thesis we have given a brief introduction to minimal submanifolds,focusing most of our attention on minimal submanifolds of Sn. In particu-lar, we gave a detailed proof of a characterization result due to Simons [16]:out of all k-dimensional minimal submanifolds of Sn, the embedded, totallygeodesic Sk are those which minimize the index. This led us to introduceConjecture 2.2.1.1 and discuss Perdomo?s [14] partial results with an addi-tional symmetry assumption. Finally, we presented Urbano?s proof of theconjecture in the special case when n = 3, and noted the tools used in theproof that prevent it from extending to higher dimensions.Since relatively few examples of minimal hypersurfaces of Sn are know,it is hard to say how strong Perdomo?s additional symmetry assumptionactually is. One possible course for further research would be to try toconstruct more examples of minimal hypersurfaces in Sn, especially ones oflow index or ones that do not satisfy Perdomo?s symmetry condition. It ispossible that this could lead one to better understand how this symmetrycondition and the index are linked. Even though the conjecture is knownto be true in the case when n = 3, it may still be worthwhile to try toconstruct minimal surfaces in this special case. Lawson [8] has proved astrong existence result for minimal surfaces in S3 that we also do not havein higher dimensions, so focusing efforts on the case when n = 3 may bemore fruitful. One could hope to extend methods used to construct minimalsurfaces in S3 to higher dimensions.Another possible direction would be to look at the analogous problemsfor the free boundary problem in Bn, the standard n-dimensional unit ballin Rn. Moving to the free boundary situation would change the problemsslightly: we would instead be considering minimal submanifolds with bound-44Chapter 3. Conclusionary in the boundary of the ball, and meeting the boundary of the ball or-thogonally. These submanifolds still arise variationally as critical points ofthe area functional, but among submanifolds in the ball whose boundaries lieon the boundary of the ball but are free to vary on the boundary. An inter-esting open problem is the analogue of Urbano?s result: every free boundaryminimal surface S ? B3 satisfies ind(S) ? 4 with equality holding if andonly if S is the critical catenoid. One could also ask whether some analo-gous form of Lawson?s existence result is true in the free boundary setting:which compact orientable surfaces with boundary can be realized as prop-erly embedded free boundary minimal surfaces in B3. A classical result ofNitsche [12] showed that the equatorial disk is the only immersed minimaldisk in B3 with free boundary, and it is conjectured that, for the case of theannulus, the critical catenoid is the unique properly embedded minimal an-nulus in B3 with free boundary. This conjecture can be viewed as an analogfor free boundary minimal surfaces in B3 of a famous conjecture of Lawsonthat asserts that the Clifford torus is the unique embedded minimal torusin S3, recently settled by Brendle. As is the case with Sn, few examples ofminimal submanifolds are known in the free boundary setting, so it wouldbe interesting to work on construction problems there as well.45Bibliography[1] F. J. Almgren Jr, Some Interior Regularity Theorems for Minimal Sur-faces and an Extension of Berstein?s Theorem, Annals of Mathematics,Second Series, 84, no. 2 (1966), 277-292.[2] M. Berger, P. Gauduchon, E. Mazet, Le Spectre d?une Varie?te? Rieman-nienne, Lecture Notes in Mathematics vol. 194, Springer-Verlag, Berlin(1971), 144.[3] Simon Brendle, Embedded Minimal Tori in S3 and the Lawson Conjec-ture, arXiv:1203.6597v2 (2012).[4] Isaac Chavel, Eigenvalues in Riemannian Geometry, Academic PressInc. Orlando, FL (1984), 1-10.[5] S. S. Chern, M. Do Carmo, and S. Kobayashi, Minimal Submanifoldsof a Sphere with Second Fundamental Form of Constant Length. Func-tional Analysis and Related Fields, Proc. Conf. M. Stone, Springer,1970, 59-75.[6] Tobias Holck Colding and William P. Minicozzi II, A Course in Min-imal Surfaces, Graduate Studies in Mathematics vol. 121, AmericanMathematical Society, Providence, RI (2011), 1-42.[7] Ju?rgen Jost, Two-Dimensional Geometric Variational Problems Wiley-Interscience, England (1991), 1-20.[8] H. Blaine Lawson Jr, Complete Minimal Surfaces in S3. The Annals ofMathematics, Second Series, 92, no. 3 (1970), 335-374.46Bibliography[9] H. Blaine Lawson Jr, Lectures on Minimal Submanifolds, Volume 1.Publish or Perish, Inc. Berkeley, CA. (1980), 1-25.[10] John M. Lee, Riemannian Manifolds: an Introduction to Curvature.Graduate Texts in Mathematics, Springer-Verlag, New York, NY(1997).[11] Sebastia?n Montiel and Antonio Ros, Minimal Immersions of Surfaces bythe First Eigenfunctions and Conformal Area, Invent. Math. 83 (1986),153-166.[12] Johannes C. C. Nitsche, Stationary Partitioning of Convex Bodies.Arch. Rat. Mech. Anal., 89, no. 1 (1985), 130-139.[13] Morio Obata, Certain Conditions for a Riemannian Manifold to beIsometric with a Sphere, J. Math. Soc. Japan, 14, no. 3 (1962), 333-340.[14] Oscar Perdomo, Low Index Minimal Hypersurfaces of Spheres, Asian J.Math. 5 no. 4 (2001), 741-750.[15] Takashi Sakai, Riemannian Geometry. Translations of MathematicalMonographs vol. 149, American Mathematical Society, Providence, RI(1996), 262-272.[16] James Simons, Minimal Varieties in Riemannian Manifolds. Annals ofMath. Second Series, 88 no. 1 (1968), 62-105.[17] Michael Taylor, Partial Differential Equations I: Basic Theory AppliedMathematical Sciences vol. 115, Springer-Verlag, New York, NY (1996),379-390.[18] Francisco Urbano, Minimal Surfaces with Low Index in the Three-dimensional Sphere, Proc. of the Amer. Math. Soc. , 108 no. 4 (1990),989-992.47
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Title | Minimal hypersurfaces of the round sphere |
Creator |
Sargent, Pamela |
Publisher | University of British Columbia |
Date Issued | 2013 |
Description | The purpose of this thesis is to discuss a conjectured classification concerning the index of non-totally geodesic minimal hypersurfaces of the n-dimensional standard sphere of radius one S^n. We briefly discuss the basic theory of minimal submanifolds before turning our attention to minimal submanifolds and hypersurfaces in S^n. We present some results of Simons which show that any minimal submanifold of S^n is unstable, and how the totally geodesic S^k ⊂ S^n are characterized by their index. We then present a related conjecture which claims that the Clifford hypersurfaces are also characterized by their index in a similar way, discuss the most recent developments related to the conjecture, and give Urbano’s proof of the conjecture for the special case when n = 3 |
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Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2013-08-12 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0074044 |
URI | http://hdl.handle.net/2429/44774 |
Degree |
Master of Science - MSc |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2013-11 |
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UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
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