Uniqueness of Lagrangian mean curvature flow and minimal immersions with free boundary by Chao Pang A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) May 2013 c Chao Pang 2013 Abstract In this thesis we investigate some problems on the uniqueness of mean curvature flow and the existence of minimal surfaces, by geometric and analytic methods. A summary of the main results is as follows. (i) The special Lagrangian submanifolds form a very important class of minimal submanifolds, which can be constructed via the method of mean curvature flow. In the graphical setting, the potential function for the Lagrangian mean curvature flow satisfies a fully nonlinear parabolic equation n ∂u = arctan λj ∂t (1) j=1 u(x, 0) = u0 (x) where the λj ’s are the eigenvalues of the Hessian D2 u. We prove a uniqueness result for unbounded solutions of (1) without any growth condition, via the method of viscosity solutions ([4], [13]): for any continuous u0 in Rn , there is a unique continuous viscosity solution to (1) in Rn × [0, ∞). (ii) Let N be a complete, homogeneously regular Riemannian manifold of dimN ≥ 3 and let M be a compact submanifold of N . Let Σ be a compact Riemann surface with boundary. A branched immersion u : (Σ, ∂Σ) → (N, M ) is a minimal surface with free boundary in M if u(Σ) has zero mean curvature and u(Σ) is orthogonal to M along u(∂Σ) ⊆ M . We study the free boundary problem for minimal immersions of compact bordered Riemann surfaces and prove that • if Σ is not a disk, then there exists a free boundary minimal immersion of Σ minimizing area in any given conjugacy class of a map in C 0 (Σ, ∂Σ; N, M ) that is incompressible; • the kernel of i∗ : π1 (M ) → π1 (N ) admits a generating set such that each member is freely homotopic to the boundary of an area minimizing disk that solves the free boundary problem. ii Abstract (iii) Under certain nonnegativity assumptions on the curvature of a 3manifold N and convexity assumptions on the boundary M = ∂N , we investigate controlling topology for free boundary minimal surfaces of low index: • We derive bounds on the genus, number of boundary components; • We prove a rigidity result; • We give area estimates in term of the scalar curvature of N . iii Preface This thesis is based on two research papers: [1] J. Chen, C. Pang, Uniqueness of unbounded solutions of Lagrangian mean curvature flow, C. R. Math. Acad. Sci. Paris 347 (2009), 1031–1034. [2] J. Chen, A. Fraser, and C. Pang, Minimal immersions of compact bordered Riemann surfaces with free boundary, to appear in Trans. Amer. Math. Soc., arXiv:1209.1165. All authors contributed equally, and are listed alphabetically, as is the convention in mathematics. iv Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Preface Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Uniqueness of Lagrangian Mean Curvature Flow . 2.1 Lagrangian mean curvature flow . . . . . . . . . . . 2.2 A comparison theorem for viscosity solutions . . . . 2.2.1 The notion of viscosity solutions . . . . . . . 2.2.2 A comparison theorem of of Barles-Biton-Ley 2.3 Hypotheses (H1) . . . . . . . . . . . . . . . . . . . . 2.3.1 First proof . . . . . . . . . . . . . . . . . . . 2.3.2 Second proof . . . . . . . . . . . . . . . . . . 2.4 Hypotheses (H2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 9 11 11 12 13 13 14 16 3 Free boundary minimal immersions . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Existence of minimizing harmonic maps in a conjugacy class 3.2.1 Existence of minimizers for Eα . . . . . . . . . . . . . 3.2.2 Euler-Lagrange equations for Eα . . . . . . . . . . . . 3.2.3 Main estimates . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Convergence of critical maps for Eα as α → 1 . . . . 3.2.5 Existence of a harmonic map in any homotopy class . 3.2.6 Convergence of harmonic maps for varying conformal structures . . . . . . . . . . . . . . . . . . . . . . . . 18 19 20 20 21 24 29 31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 37 v Table of Contents 3.3 A variational problem on the space of conformal structures M for compact Riemann surfaces . . . . . . . . . . . . . . . 3.3.1 The notion of Riemann moduli and Teichm¨ uller spaces 3.3.2 Structure theorem for M . . . . . . . . . . . . . . . . 3.3.3 Variational problems of conformal structures . . . . . ¯ 3.3.4 Minimal area problem, 1st approach: the functional E on M . . . . . . . . . . . . . . . . . . . . . . . . . . . ¯ 3.3.5 Minimal area problem, 2nd approach: the functional E on the Teichm¨ uller space . . . . . . . . . . . . . . . . Minimal surfaces of non-disk type . . . . . . . . . . . . . . . 3.4.1 First proof . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Second proof . . . . . . . . . . . . . . . . . . . . . . . Minimizing disks . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 An integral identity for critical maps of Eα . . . . . . 3.5.2 Nontrivial limit of critical maps of Eα . . . . . . . . . 3.5.3 Minimizing disks as generators . . . . . . . . . . . . . Topology of minimal surfaces of low index . . . . . . . . . . 3.6.1 The first eigenfunction of the index form . . . . . . . 3.6.2 A stability inequality for index-1 minimal surfaces . . 3.6.3 Controlling topology, area estimates, and rigidity for minimal surfaces of low index . . . . . . . . . . . . . 64 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.4 3.5 3.6 39 39 41 43 45 49 50 50 52 54 55 56 58 62 62 63 vi Acknowledgements I am grateful to my supervisors, Professor Jingyi Chen and Professor Ailana Fraser, for giving me an opportunity of studying at UBC and introducing a beautiful field to me, and for their constant support and encouragement. I am appreciative for their knowledge and professional skills, from which I have never stopped learning, and to their broad perspectives in and outside mathematics. I am grateful to Professor Tobias Lamm (Karlsruhe Institute of Technology) for his help and exemplary power. Also to the wonderful staff in the mathematics department, many colleagues and friends at UBC and from conferences. In Canada I have been influenced by discoveries in my reading and thinking that have revolutionized my mind and initialized new thoughts. I have been inspired by many people I encountered, and many of the novel perspectives the country has provided. These differences pave the path for integration. I am appreciative for the hospitable, broad, and free spirits in this amazing land. To my friends: Ates Tanin (Toronto, Canada) Davide Ciaccia (Parma, Italy) Stefano Lania (Bergamo, Italy) Werner Unger (Kehl/Rhein, Germany) vii Dedication Through the Ph.D. process at UBC and at all times, I was accompanied by my Aunt’s unconditional warmth, love and devotion. To the memory of Hibiscus mutabilis. viii Chapter 1 Introduction The field of minimal surfaces has its origin in the mid eighteenth century with the publication of Lagrange’s famous memoir “Essai d’une nouvelle m´ethode pour d´eterminer les maxima et les minima des formules int´egrales ind´efinies” and Euler’s paper on the minimizing properties of the catenoid. The study of minimal surfaces has remained a vibrant area of research since Lagrange’s memoir. A generalization of the concept of a minimal surface to higher dimen¯ n+k be an isometrically sions is a minimal submanifold. Let F0 : M n → M immersed submanifold. A minimal submanifold is a critical point of the ¯ be a family of immersions such volume functional. Let F : (− , ) × M → M ∂ that F (0, ·) = F0 . Denote by X = (F∗ ∂t )(0, ·) the variational vector field. The first variation formula says that M is minimal if d Vol(Ft (M )) dt t=0 =− X, H dσ = 0 M where H is mean curvature vector. Thus a submanifold is minimal if its mean curvature vector vanishes. In an orthonormal basis {ei } for the tangent space at one point, H is given by H = A(ei , ei ), where A is the second fundamental form. The field of minimal submanifolds has been a central area of research and is closely related to other branches of mathematics and mathematical physics such as topology, partial differential equations and mathematical relativity. It has been an active research field which has given rise to a wide variety of problems, which led to the discovery of many profound phenomena in mathematics. However, the understanding of minimal submanifolds is still far from being complete even in dimension two, i.e. in the case of minimal surfaces; and there are many open problems even in one of the simplest cases of embedded minimal surfaces in the Euclidean 3-sphere. For example, one of the many conjectures concerning embedded minimal surfaces in the 3-sphere, the so-called Lawson conjecture, was only recently solved by S. Brendle. 1 Chapter 1. Introduction A special class of minimal submanifolds arises from calibrated geometry (cf. [25]). Let X be a Riemannian manifold. A calibration on X is a closed p-form ϕ that satisfies ϕ(σ) ≤ Vol(σ) (1) for all oriented tangent p-planes σ on X, and X is called a calibrated manifold. The oriented p-planes at which equality is achieved are called ϕ-planes. A ϕ-submanifold is a compact oriented p-dimensional submanifold M of X such that the tangent space at each point is a ϕ-plane. This implies ϕ(M ) = Vol(M ). (2) It is a fundamental fact of calibrated geometry that ϕ-submanifolds are homologically volume minimizing (since for any M with ∂M = ∂M which is homological to M , by (1), (2) and since ϕ is a closed form, we have Vol(M ) = M ϕ = M ϕ ≤Vol(M )). In particular, ϕ-submanifolds have vanishing mean curvature vectors. Many interesting examples of calibrated manifolds exist, such as K¨ahler, Calabi-Yau, G2 , or spin(7) manifolds. One of the calibrated geometries of primary importance is associated to the form ϕ = Re(dz1 ∧ . . . ∧ dzn ) in Cn = Rn ⊕ iRn . The ϕ-submanifolds consist of Lagrangian submanifolds of Cn which are stationary, and are therefore called special Lagrangian submanifolds. Since all Lagrangian planes in Cn are equivalent under SUn to the real plane Rn = (Rn , 0) ⊂ Cn , we may consider a special Lagrangian submanifold M to be given locally as a graph (x, F (x)) over a domain in Rn . Then the graph is Lagrangian if and only if the Jacobian matrix (∂F i /∂xj ) is symmetric. In particular, if the domain is simple-connected (for example an entire graph) the Lagrangian condition implies that locally F = ∇f for some potential function. In order for the graph to be special Lagrangian it 2f must be Lagrangian and satisfy one other condition. Let Hess f = ∂x∂i ∂x j denote the Hessian of f . Then it is special Lagrangian if and only if f satisfies a fully nonlinear elliptic equation Im{detC (I + iHess f )} = 0. In particular, this equation reads ∆f = det(Hess f ) in dimension three. That is the Laplacian of f equals the Monge-Amp´ere of f . 2 Chapter 1. Introduction The existence of minimal submanifolds in Rn or general Riemannian manifolds is a fundamental topic in the theory of minimal submanifolds. One approach that has been proposed for constructing minimal submanifolds is via the parabolic method, i.e. the mean curvature flow. Namely, given an embedding F0 : M → N , the mean curvature flow is a family of maps F : M × [0, T ) → N that satisfy ∂F ∂t =H F (x, 0) = F0 (x). A stationary solution, if one exists, will be a minimal submanifold. In the Lagrangian setting, the special Lagrangian mean curvature flow equation for graphs takes the form ∂f ∂t = √1 −1 √ −1D2 f ) det(In +(D2 f )2 √ n+ log det(I (3) f (x, 0) = f0 (x) which is a fully nonlinear parabolic equation. Suppose a solution exists. Then a family of diffeomorphisms φt : Rn → Rn can be constructed accordingly such that F (x, t) = (φt (x), ∇f (φt (x), t)) evolves by the mean curvature flow equation ∂F ∂t = H F (x, 0) = ∇f0 (x). A stationary solution will be minimal. The long time existence for solutions to (3) with C 1,1 initial data and a certain bound on the Hessian has been established by Chau-Chen-He [6] and Chau-Chen-Yuan [8]. The solutions become smooth immediately. For less smooth initial data, to study the problem one has to use other weaker solutions. Since the differential operator is in the Hessian form, a natural way to study the problem is by viscosity solutions. In Chapter 2, we prove the existence of a viscosity solution to (3) for any continuous initial data. On the other hand, uniqueness results can also find many applications. We also investigate the uniqueness problem for (3) and prove the following result (a similar result holds for the Cauchy-Dirichlet problem). Theorem 1.0.1. Let u and v be an upper semicontinuous and a lower semicontinuous viscosity subsolution and supersolution to (3) in Rn × [0, T ) respectively. If u(x, 0) ≤ v(x, 0) for all x ∈ Rn , then u ≤ v in Rn × [0, T ). In particular, for any continuous function u0 in Rn , there is a unique continuous viscosity solution to (3) in Rn × [0, ∞). 3 Chapter 1. Introduction From a purely analytic point of view, when the domain is noncompact, it is often necessary to impose certain growth conditions on the solutions at infinity in order to obtain uniqueness for nonlinear parabolic equations. In our proof, no growth condition is assumed. Instead, a certain boundedness of the differential operator is crucial in applying a comparison theorem of Barles-Biton-Ley ([4]) for viscosity solutions of a class of fully nonlinear parabolic equations. This result has been used by Chau-Chen-He ([7]) in studying the rigidity of self-similar solutions of the Lagrangian mean curvature flow (see also [9]). There is another way of proving the existence of minimal submanifolds, which can be formulated into a variational problem of certain functionals in various settings. In this approach, the Euler-Lagrange equation of the functional plays an important role, which is an elliptic differential equation, and therefore we may call this approach an elliptic method. The boundary value problem for minimal surfaces began with the study of Plateau’s problem, which is the question of finding a surface of least area spanning a fixed simple Jordan curve. A general solution was obtained by Douglas and simultaneously by Rad´o. For minimal surfaces in Riemannian manifolds, the problem was solved by Morrey [45], and was later supplemented by results of Lemaire [36], and Jost [29]. A general existence result for minimal immersions of closed Riemann surfaces of genus ≥ 1 in compact manifolds was proved independently by Schoen and Yau [50] (using the energy E(f ) = Σ |∇f |2 dµg ), and Sacks and Uhlenbeck [48, 49] (using perturbed energy functionals Eα (f ) = Σ (1 + |∇f |2 )α dµg ). Their theorems show that for any continuous map f : Σ → N such that f∗ : π1 (Σ) → π1 (N ) is injective, there exists a branched minimal immersion of Σ minimizing area among all maps which induce the same action on the fundamental group. Sacks and Uhlenbeck [48] also established an existence theory for minimal 2-spheres. In particular, they proved that if the universal covering space of N is not contractible, then there exists a non-trivial branched minimal immersion of the 2-sphere into N . Another natural boundary value problem is the free boundary problem, where the boundary of a surface is allowed to vary in some supporting submanifold. Let M be a closed submanifold of a Riemannian manifold N . A branched immersion u : (Σ, ∂Σ) → (N, M ) of a surface Σ with nonempty boundary ∂Σ is a minimal surface with free boundary in M if u(Σ) has zero mean curvature and u(Σ) is orthogonal to M along u(∂Σ) ⊆ M . Existence results for free boundary minimal surfaces in various settings have been much studied. 4 Chapter 1. Introduction In Chapter 3 we study the free boundary problem for minimal immersions of compact bordered Riemann surfaces. The purpose is twofold. In the first part, we prove a general existence theorem for compact bordered Riemann surfaces of any topological type in complete Riemannian manifolds, assuming certain incompressibility conditions. In the second part, we investigate controlling the topology of free boundary minimal surfaces of low index in 3-manifolds, under certain nonnegativity assumptions on the curvature and convexity assumptions on the boundary of the 3-manifold. Our existence result is: Theorem 1.0.2. Let N be a complete, homogeneously regular Riemannian manifold of dimN ≥ 3 and let M be a compact submanifold of N . Then, (i) if Σ is a compact, connected orientable surface of genus g and with k ≥ 1 boundary components that is not a disk, and f : Σ → N is a continuous map with f (∂Σ) ⊂ M such that f∗ : π1 (Σ) × π1 (Σ, ∂Σ) → π1 (N ) × π1 (N, M ) is injective, then there exists a branched minimal immersion (Σ, ∂Σ) → (N, M ) solving the free boundary problem, and minimizing area among all maps (Σ, ∂Σ) → (N, M ) that induce the same action as f on the fundamental groups; (ii) there exists a generating set {γj } for ker i∗ , where i∗ : π1 (M ) → π1 (N ) is the homomorphism induced by the inclusion, such that each γj is freely homotopic to the boundary of an area minimizing disk that solves the free boundary problem. The disk case, part (ii) of the above theorem, was already proved by Ye [57] by a different method. Existence results for disk-type solutions in various settings have been studied by Meeks and Yau [44], Jost [30–32], Struwe [54], Kuwert [35], Fraser [22], among others. Embedded free boundary solutions of prescribed topological type in 3-manifolds with mean convex boundary were produced by Jost [31] assuming a Douglas type condition. Recently M. Li [37] proved existence of embedded solutions of controlled topological type in 3-manifolds with no convexity assumption on the boundary. 5 Chapter 1. Introduction We take the Sacks-Uhlenbeck approach of working with the perturbed energy. The analytic foundation is already established in the interior [48] and in the boundary [22] settings. Following the ideas of Schoen-Yau [50] and Sacks-Uhlenbeck [49] for closed surfaces, for each conformal structure on the bordered surface we produce an energy-minimizing map which induces the same action on the fundamental group as a given continuous map f : (Σ, ∂Σ) → (N, M ), and then we minimize over all conformal structures to produce a branched minimal surface. The key point is to understand the limiting of the conformal structures in the boundary setting. The incompressibility assumptions on the fundamental groups prevent degeneration in the limiting of the conformal structures, in the form of pinching of an interior or boundary circle or of a curve connecting two boundary circles of the surface. Remark. In the special case that M is 1-dimensional, the theorem asserts a solution to the Plateau problem, i.e. an area-minimizing immersion of Σ with fixed boundary as M in N . Besides preventing degeneration of a sequence of conformal structures on Σ, the incompressibility assumptions are also used to rule out triviality of the energy-minimizing maps for each fixed conformal structure. When the ambient manifold has positive curvature, there are some strong restrictions on the topology of stable or index one minimal surfaces. In the second part of the paper, we investigate the relationship between the topology of free boundary minimal surfaces and the geometry of the ambient manifold, such as nonnegativity of the curvature and convexity of the boundary, by means of the second variation formula. One of the reasons we are interested in understanding the topology of minimal surfaces in a Riemannian manifold is because it provides information about the ambient manifold (e.g. see [56]). For closed minimal surfaces in 3-manifolds of positive Ricci curvature, it is conjectured that there should exist a bound on the genus of any minimal surface in terms of its Morse index, and it is known that any index 1 surface must have genus at most three ([47]). For minimal surfaces with free boundary in 3-manifolds with nonnegative Ricci curvature and weakly convex boundary, we obtain a bound on the genus and number of boundary components of any index 1 free boundary minimal surface. Also, there is a bound on the area of stable or index 1 free boundary solutions in terms of the topology of the surface and a positive lower bound on the ambient scalar curvature. 6 Chapter 1. Introduction Now we state the theorem. Theorem 1.0.3. Let N be a 3-dimensional Riemannian manifold with smooth boundary ∂N . Suppose Σ is a compact orientable two-sided surface of genus g and with k ≥ 1 boundary components, solving the free boundary problem (Σ, ∂Σ) → (N, ∂N ). (i) Suppose Ric(N ) ≥ 0 and ∂N is weakly convex. If Σ has index 1, then: a) g + k ≤ 3 if g is even; b) g + k ≤ 4 if g is odd. (ii) Suppose the scalar curvature R(N ) ≥ 0 and ∂N is weakly mean convex. If Σ is stable, then Σ is either a disk or a totally geodesic and flat cylinder. If Σ has index 1, then: a) k ≤ 5 if g is even; b) k ≤ 7 if g is odd. (iii) Suppose N has scalar curvature R ≥ R0 > 0 and ∂N is weakly mean convex. a) if Σ is stable, then it is a disk and Area(Σ) ≤ b) if Σ has index 1, then Area(Σ) ≤ 2π R0 ; 2π(7−(−1)g −k) . R0 M. Li [38] proved an area bound and rigidity result for free boundary minimal surfaces in strictly convex domains in R3 . These area estimates are free boundary analogs of the area estimates for closed stable and index 1 minimal surfaces in 3-manifolds of positive scalar curvature of Marques and Neves [42]. The rigidity result for stable surfaces in part (ii) can be viewed as the free boundary analog of results of Schoen and Yau for compact ambient manifolds ([50] Theorem 5.1), and Fischer-Colbrie and Schoen for complete ambient manifolds ([21] Theorem 3). 7 Chapter 2 Uniqueness of unbounded solutions of Lagrangian mean curvature flow for graphs 8 2.1. Lagrangian mean curvature flow 2.1 Lagrangian mean curvature flow The special Lagrangian graphs equation was derived in Harvey-Lawson [25]. Suppose F : Rn → Cn = Rn ⊕ iRn is a C 1 -mapping. Then the graph of F is Lagrangian if and only if F = ∇f for some potential function f in C 2 (Rn ). In addition, the graph (x, ∇f ) is a special Lagrangian submanifold in Cn if and only if f satisfies the following differential equation Im detC In + iD2 f = 0. Since this condition is equivalent to det In + iD2 f = det In + iD2 f = det In − iD2 f , we have i log det In + iD2 f =0 det In − iD2 f (mod 2π). The parabolic version of this equation, i.e. the Lagrangian mean curvature flow equation, thus takes the form ∂f ∂t = det In + iD2 f 1 log 2i det In − iD2 f = 1 log i det(In + iD2 f ) det(In + (D2 f )2 where f is a function Rn × [0, t) → R2n . The factor 1i is introduced for the right hand side to take real values. Indeed denote by λj ’s the eigenvalues of D2 f . We have det(In + iD2 f ) = (1 + iλj ) = 1 (1 + λ2j ) 2 · ei arctan λj and then det In + iD2 f 1 log 2i det In − iD2 f = 1 ei log −i 2i e arctan λj arctan λj n = arctan λj (mod 2π). j=1 Hence we can write the equation also as ∂f = ∂t n arctan λj (mod π). j=1 9 2.1. Lagrangian mean curvature flow Accordingly the initial value problem can be formulated as the following n ∂u = arctan λj ∂t (2.1.1) j=1 u(x, 0) = u0 (x) for u0 : Rn → R a given function and u : Rn × [0, T ) → R. For each t, (x, ∇u(x)) defines a Lagrangian submanifold in R2n . In particular, for a smooth stationary solution to (2.1.1), the graph of its gradient is a special Lagrangian submanifold in R2n , which can be seen from det In + iD2 f =1 det In − iD2 f (see also [53], [52]). Suppose u is a regular solution to (2.1.1). Then a family of diffeomorphisms φt : Rn → Rn can be constructed such that F (x, t) = (φt (x), ∇u(φt (x), t)) solves the mean curvature flow equation ∂F ∂t =H F (x, 0) = ∇f0 (x), and so we call a solution to 2.1.1 the Lagrangian mean curvature flow. The smooth longtime existence of solutions to (2.1.1) has been established in Chau-Chen-He [6] and Chau-Chen-Yuan [8] with C 1,1 initial u0 and assuming a certain bound on the Lipschitz norm of Du0 . In this chapter, we study uniqueness problems for the Lagrangian mean curvature flow. First, we consider (2.1.1) on unbounded domain Rn . The result is Theorem 2.1.1. Let u and v be an upper semicontinuous and a lower semicontinuous viscosity subsolution and supersolution to (2.1.1) in Rn × [0, T ) respectively. If u(x, 0) ≤ v(x, 0) for all x ∈ Rn , then u ≤ v in Rn × [0, T ). In particular, for any continuous function u0 in Rn , there is a unique continuous viscosity solution to (2.1.1) in Rn × [0, ∞). Then we also consider the Cauchy-Dirichlet problem for the Lagrangian mean curvature flow. The key ingredient in the proof is a comparison theorem of Barles-Biton-Ley for a class of parabolic equations. 10 2.2. A comparison theorem for viscosity solutions 2.2 A comparison theorem for viscosity solutions 2.2.1 The notion of viscosity solutions The theory of viscosity solutions provides a framework for proving comparison and uniqueness theorems, and existence theorems for fully nonlinear partial differential equations of second order, which have found enormous important applications (cf. Crandall-Iishi-Lions [13]). Suppose an elliptic equation is in the form F (x, u, Du, D2 u) = 0, where F : Rn × R × Rn × Sn → R and Sn is the space of symmetric n × n matrices, and F satisfies a monotonicity condition F (x, r, p, X) ≤ F (x, s, p, Y ), wherever r ≤ s and Y ≤ X where Sn is equipped with its usual order, i.e. we say X ≥ Y if and only if (Xe, e) ≥ (Y e, e), ∀e ∈ Rn . Let USC (LSC ) denote the set of upper (lower) semicontinuous functions on Ω ⊂ Rn . For u : Ω → R and x ∈ Ω, let JΩ2,+ u(x), the second-order superject of u at x, be the set of (p, X) ∈ Rn × Sn such that u(x ) ≤ u(x) + p · (x − x) + X(x − x) · (x − x)/2 + o |x − x|2 for any x → x. Definition 2.2.1 ([13]). A viscosity subsolution (supersolution) of F = 0 on Ω is a function u ∈ USC (LSC) such that F x, u(x), p, X ≤ (≥) 0, ∀x ∈ Ω, (p, X) ∈ JΩ2,+ u(x) − JΩ2,− u(x) . A viscosity solution of F = 0 is both a viscosity subsolution and a viscosity supersolution. With certain conditions on the differential operator F , a maximal principle and a comparison result for viscosity solutions can be established ([13]). Furthermore, the definition of viscosity solutions and the comparison result can be extended to parabolic equations ut + F t, x, u, Du, D2 u = 0 where u is a function u(t, x), Du = Dx u(t, x), D2 u = Dx2 u(t, x) and F : R × Rn × R × Rn × Sn → R ([13]). 11 2.2. A comparison theorem for viscosity solutions 2.2.2 A comparison theorem of of Barles-Biton-Ley Barles, Biton and Ley obtained a general comparison result (Theorem 2.1 in [4]) for the viscosity solutions of a class of fully nonlinear parabolic equations, as well as an existence result (Theorem 3.1 in [4]). We now describe the assumptions in the comparison and existence results in [4]. For any X ∈ Sn , there exists an orthogonal matrix P such that X = P ΛP T , where Λ is the diagonal matrix with diagonal entries consisting of eigenvalues of X. Let Λ+ be the diagonal matrix obtained by replacing the negative eigenvalues in Λ with 0’s. Then we define X + = P Λ+ P T . Suppose F is a continuous function from Rn × [0, T ] × Rn × Sn to R. Consider the following assumptions on F ([4]): (H1) For any R > 0, there exists a function mR : R+ → R+ such that mR (0+ ) = 0 and F (y, t, η(x − y), Y ) − F (x, t, η(x − y), X) ≤ mR (η |x − y|2 + |x − y|) ¯ R (0) and t ∈ [0, T ], whenever X, Y ∈ Sn and η > 0 satisfy for all x, y ∈ B −3η I 0 0 I ≤ X 0 0 −Y ≤ 3η I −I −I I (H2) There exist 0 < α < 1 and constants K1 > 0 and K2 > 0 such that F (x, t, p, X) − F (x, t, q, Y ) ≤ K1 |p − q| (1 + |x|) + K2 tr (Y − X)+ α for every (x, t, p, q, X, Y ) ∈ Rn × [0, T ] × Rn × Rn × Sn × Sn . The operator F is degenerate elliptic if (H2) holds. Theorem 2.2.2. (Barles-Biton-Ley) Let u and v be an upper semicontinuous viscosity subsolution and a lower semicontinuous viscosity supersolution respectively of ∂u + F (x, t, Du, D2 u) = 0 in Rn × [0, T ) ∂t u(·, 0) = u0 in Rn . Assume that (H1) and (H2) hold for F . Then (i) If u(·, 0) ≤ v(·, 0) in Rn , then u ≤ v in Rn × [0, T ). (ii) If u0 ∈ C(Rn ), there is a unique continuous viscosity solution in Rn × [0, ∞). 12 2.3. Hypotheses (H1) In particular, they showed that (2.1.1) admits a unique longtime continuous viscosity solution for any continuous function u0 in R when n = 1. In the next section, we observe, via elementary methods, that the hypotheses in the general theorems in [4] are valid for the geometric evolution equation (2.1.1) in general dimensions. 2.3 Hypotheses (H1) We now present the proof of Theorem 2.1.1. We define F : Sn → R by F (X) = −i log det(I + iX) i det(I + iX) = − log . 2 det(I − iX) det(I + X 2 ) 1 2 (2.3.1) That F takes real values follows easily from F (X) = i det(I − iX) log = F (X). 2 det(I + iX) Note that F (D2 u) is equal to arctan λj modulo 2π (see Section 2.1). Therefore the flow (2.1.1) is equivalent to ut + (−F (D2 u)) = 0. Since F (x, t, p, X) = F (X) is independent of x, the right hand side of the inequality for F in (H1) must be zero, namely mR = 0. By multiplying an arbitrary vector (ξ, ξ) ∈ Rn × Rn and its transpose to the second matrix inequality in (H1), we see that X ≤ Y . Therefore, in order to establish (H1) it suffices to show: (H1 ) For any X, Y ∈ Sn , if X ≥ Y then F (X) ≥ F (Y ). 2.3.1 First proof For any X, Y ∈ Sn and t ∈ [0, 1], define fXY (t) = F (tX + (1 − t)Y ). We will show that fXY (t) is nondecreasing in t ∈ [0, 1] and then (H1 ) will follow as fXY (0) = F (Y ) and fXY (1) = F (X). Set A = I + i(tX + (1 − t)Y ) and B = I − i(tX + (1 − t)Y ). 13 2.3. Hypotheses (H1) Then i fXY (t) = − (log det A − log det B). 2 It follows that AB = BA and A−1 + B −1 · A+B AB = = I. 2 2 Note that both A and B are invertible matrices for all t ∈ [0, 1]. Hence, by using the formula ∂t ln det G = tr(G−1 ∂t G) for G(t) ∈ GL(n, R), we have i fXY (t) = − tr A−1 · ∂t A − B −1 · ∂t B 2 i = − tr (A−1 + B −1 ) · i(X − Y ) 2 i = − tr (A + B)(AB)−1 · i(X − Y ) 2 = tr (I + (tX + (1 − t)Y )2 )−1 · (X − Y ) . (2.3.2) Since tX + (1 − t)Y is real symmetric, the matrix C = I + (tX + (1 − t)Y )2 is positive definite, hence so is C −1 . There exists a matrix Q ∈ GL(n, R) such that C = QQT . By the assumption X ≥ Y , we have tr C −1 (X − Y ) = tr Q · QT (X − Y ) = tr QT (X − Y ) · Q ≥ 0 since QT (X − Y )Q is positive semidefinite. Therefore, we have shown that (H1) is valid for F defined in (2.3.1). 2.3.2 Second proof We notice that (H1 ), for the operator F (X) = arctan λj (X), also follows from the basic fact: Suppose X, Y ∈ Sn and that the eigenvalues λj of X and µj of Y are in descending order λ1 ≥ λ2 ≥ · · · ≥ λn and µ1 ≥ µ2 ≥ · · · ≥ µn . If X ≥ Y , then λj ≥ µj for j = 1, · · · , n (see p.182 in [27]). Here we include a proof. Proof. Let ei be an eigenvector to λi such that {ei } form an orthogonal basis. Denote Vk = span{e1 , · · · , ek } and Vk⊥ = span{ek+1 , · · · , en } its 14 2.3. Hypotheses (H1) orthogonal complement. Also let hi be an eigenvector to µi and denote Wk = span{h1 , · · · , hk } and Wk⊥ = span{hk+1 , · · · , hn }. By the variational characterization and the descending order of eigenvalues, we have λk = max ⊥ e∈Vk−1 (Xe, e) , e 2 µk−1 = min h∈Wk−1 (Y h, h) . h 2 ⊥ ∩W Now we look at the intersection Vk−1 k−1 : ⊥ ∩ W i) Vk−1 k−1 contains an unit vector e = h: then λk ≥ (Xe, e) ≥ (Y h, h) ≥ µk−1 ≥ µk , where the first inequality is by the variational ⊥ , the second is by X ≥ Y , and the third is by the formula and e ∈ Vk−1 variational formula and h ∈ Wk−1 . ⊥ ⊥ ∩W ii) Vk−1 k−1 = {0}: then V = Vk−1 ⊕ Wk−1 (the sum may not be ⊥ , there exist unit orthogonal). This implies that for any h⊥ ∈ Wk−1 ⊥ such that h⊥ − ah = be⊥ . Since h vectors h ∈ Wk−1 and e⊥ ∈ Vk−1 and h⊥ are orthogonal (with respect to the Euclidean form), we get (h⊥ , h⊥ ) + a2 = b2 . ⊥ On the other hand, since Wk−1 and Wk−1 are orthogonal also with respect to the bilinear form Y , we get b2 (Y e⊥ , e⊥ ) = a2 (Y h, h) + (Y h⊥ , h⊥ ). From these two equalities we have (Y h⊥ , h⊥ ) (h⊥ , h⊥ ) = ≤ ≤ b2 (Y e⊥ , e⊥ ) − a2 (Y h, h) b2 − a2 b2 (Xe⊥ , e⊥ ) − a2 µk−1 b2 − a2 b2 λk − a2 µk−1 b2 − a2 where the first inequality is by X ≥ Y and the variational formula, and the second inequality is by the variational formula. Taking the ⊥ , we have (b2 − a2 )µ ≤ b2 λ − a2 µ sup over Wk−1 k k k−1 . From this we derive λk ≥ µk . We have proved in either case, λj ≥ µj for j = 1, · · · , n. 15 2.4. Hypotheses (H2) 2.4 Hypotheses (H2) As F (x, t, p, X) is independent of p, (H2) reads: there exist constants K > 0 α and 0 < α < 1 such that F (X)−F (Y ) ≤ K (tr(X − Y )+ ) for all X, Y ∈ Sn . For any X, Y ∈ Sn , integrating (2.3.2) leads to 1 F (X) − F (Y ) = tr C −1 (X − Y ) dt. (2.4.1) 0 For X − Y ∈ Sn there exists an orthogonal matrix P such that X − Y = P ΛP T where the diagonal matrix Λ has diagonal entries λ1 , · · · , λn . Let −1 ≤ I, we have 0 < P T C −1 P ≤ I. If λ+ j = max{λj , 0}. Since 0 < C cjj denote the diagonal entries of P T C −1 P for j = 1, · · · , n, then cjj = P T C −1 P ej , ej where {e1 , · · · , en } is the standard basis for Rn and ·, · is the Euclidean inner product. It follows that 0 < cjj ≤ 1 for j = 1, · · · , n. Then tr C −1 (X − Y ) = tr P T C −1 P · P T (X − Y )P = tr P T C −1 P · Λ = cjj λj ≤ λ+ j = tr(X − Y )+ . Substituting the above inequality into (2.4.1) implies: for any X, Y ∈ Sn we have F (X) − F (Y ) ≤ tr (X − Y )+ . Because arctan x is in (−π/2, π/2), we have F (X) − F (Y ) < nπ. For any constant α with 0 < α < 1, (i) if tr (X − Y )+ ≤ 1, then F (X) − F (Y ) ≤ tr (X − Y )+ ≤ nπ tr (X − Y )+ α (ii) if tr (X − Y )+ > 1, then F (X) − F (Y ) ≤ nπ ≤ nπ tr (X − Y )+ α . Therefore, (H2) holds for K2 = nπ and any constants K1 > 0 and α with 0 < α < 1. 16 2.4. Hypotheses (H2) Now Theorem 2.1.1 follows immediately from Theorem 2.2.2. We also mention the uniqueness of viscosity solutions of the CauchyDirichlet problem for (2.1.1). Since the operator F (X) = arctan λj (X) satisfies (H1 ), which is exactly the fundamental monotonicity condition for −F in [13], −F is proper in the sense of [13] (p.2 [13]). As (H1) holds, Theorem 8.2 in [13] is valid for (2.1.1): Theorem 2.4.1. The continuous viscosity solution to the following CauchyDirichlet problem is unique: n arctan λj , in (0, T ) × Ω ut = j=1 u(t, x) = 0, for 0 ≤ t < T and x ∈ ∂Ω u(0, x) = ψ(x), for x ∈ Ω where λj , j = 1, · · · , n, are the eigenvalues of D2 u, Ω ⊂ Rn is open and bounded and T > 0 and ψ ∈ C(Ω). If u is an upper semicontinuous viscosity solution and v is a lower semicontinuous viscosity solution of the CauchyDirichlet problem, then u ≤ v on [0, T ) × Ω. Note that the initial boundary conditions for the subsolution and supersolution are: u(x, t) ≤ 0 ≤ v(x, t) for t ∈ [0, T ) and x ∈ ∂Ω, and u(x, 0) ≤ ψ(x) ≤ v(v, 0) for x ∈ Ω. 17 Chapter 3 Minimal immersions of compact bordered Riemann surfaces with free boundary 18 3.1. Introduction 3.1 Introduction In this chapter, we study the existence theory and second variation theory for free boundary minimal surfaces. Suppose N is a complete Riemannian manifold of dim(N ) ≥ 3, M is a compact submanifold of N , and Σ is a compact Riemann surface with nonempty boundary ∂Σ. A minimal surface of N with free boundary in M is an immersion u : (Σ, ∂Σ) → (N, M ) (possibly with branched points) such that u(Σ) has zero mean curvature and u(Σ) is orthogonal to M along u(∂Σ) ⊆ M . We begin by reviewing some basic facts about the induced homomorphism on the fundamental group and recall the Euler-Lagrange equations for the α-energy. Then we cite the homogeneously regular condition to be used in our setting, and give a detailed proof of the -regularity theorem first proved by Sacks-Uhlenbeck for closed surfaces ([48]) and then extended to the boundary situation by Fraser ([22]). Using these main estimates we obtain global convergence of critical maps of the α-energy, and the results are compiled in Subsection 3.2.4. In particular, we prove the existence of a harmonic map in any conjugacy class. Furthermore, under stronger conditions on the topology of N and M , we prove the existence of a harmonic map in any homotopy class. There we give two proofs, the first of which is based on a construction by Sacks-Uhlenbeck ([48]), and the second is based on a topological fact. Then we recall the notion of the Riemann moduli space and the Teichm¨ uller space, and a structure theorem on the space of conformal structures for compact bordered Riemann surfaces. We show how to reduce the minimal area problem to a convergence problem for conformal structures in the moduli space. Then using the incompressible assumptions, we prove a general existence theorem for free boundary minimal immersions of compact bordered Riemann surfaces which are not a disk. Two different proofs are presented in Section 3.4. Next we consider minimizing disks. We prove the kernel of i∗ : π1 (M ) → π1 (N ) admits a generating set such that each member is freely homotopic to the boundary of an area minimizing disk that solves the free boundary problem (D, ∂D) → (N, M ). In the second part, we investigate controlling the topology of free boundary minimal surfaces of low index in 3-manifolds, under certain nonnegativity assumptions on the curvature and convexity assumptions on the boundary of the 3-manifold. We derive bounds on the genus and number of boundary components, and area estimates. We also prove a rigidity result for stable minimal surfaces. 19 3.2. Existence of minimizing harmonic maps in a conjugacy class 3.2 Existence of minimizing harmonic maps in a conjugacy class Throughout this chapter the terms “fundamental group” and “relative fundamental group” will implicitly refer to some base point ∗. Since the isomorphism class of the fundamental group of a path-connected space is not affected by the choice of base-point, without ambiguity, we will write π1 ( · ) and π1 (· , ·) while omitting the base point. Any continuous map f gives rise to a homomorphism f∗ : π1 ( · , ∗) → π1 ( · , f (∗)), which we will call the induced map of f . We shall say two maps f and g induce the same action on the fundamental group, if there exists a path λ from f (∗) to g(∗), such that f∗ = λ−1 ∗ ◦ g∗ ◦ λ∗ , or equivalently, if f∗ and g∗ represent the same homomorphism after identifying the fundamental groups with different base-points through an isomorphism Iλ : π1 ( · , f (∗)) → π1 ( · , g(∗)), σ → λ · σ · λ−1 . In this case, we will briefly say f∗ is conjugate to g∗ and will write f∗ ∼ g∗ for short. Finally we recall that the relative fundamental group πn (X, A, ∗) for a triple {∗} ⊂ A ⊂ X, is a group only for n ≥ 2. When n = 1, it is the set of homotopy classes of paths from the base point ∗ to a varying point in A. Let N , M and Σ be as defined in Theorem 1.1. Given a continuous map f : Σ → N with f (∂Σ) ⊆ M , denote by f∗ the induced homomorphism as indicated in each of the following situations: 1) Σ is not a disk, 2) Σ is a disk D, f∗ : π1 (Σ) × π1 (Σ, ∂Σ) → π1 (N ) × π1 (N, M ) f∗ : π1 (∂D) → π1 (M ). We will use the terminology “the conjugacy class of f∗ ” to denote the set of maps for which the induced homomorphisms on the above fundamental groups are conjugate to f∗ . 3.2.1 Existence of minimizers for Eα Suppose a conformal structure on Σ is fixed, a Riemannian metric compatible with this conformal structure is given, and this metric defines an area element dµ. Let N → RK be a C ∞ isometric embedding for sufficiently large K. Set W 1,p (Σ, N ) = {u ∈ W 1,p (Σ, RK ) | u(x) ∈ N a.e. x ∈ Σ}. 20 3.2. Existence of minimizing harmonic maps in a conjugacy class For α > 1, we define the α-energy 1 + |∇u|2 Eα (u) = α dµ Σ on the admissible space Wα = {u ∈ W 1,2α (Σ, N ) | u(∂Σ) ⊆ M, u∗ ∼ f∗ }. Note that by the Sobolev embedding theorem, each u in W 1,2α (Σ, N ) is continuous. Proposition 3.2.1. Eα attains the infimum at some uα ∈ Wα , ∀α > 1. Proof. Let Iα = inf Eα . Let {uαk } be an Eα -minimizing sequence of maps; Wα that is Eα (uαk ) → Iα . From the Sobolev embedding W 1,2α (Σ, N ) → C 0, α−1 α (Σ, N ), the sequence {uαk } is equicontinuous, so the Arzel`a-Ascoli theorem yields a subsequence, which we still denote by {uαk }, that converges uniformly to a map uα in C 0,β (Σ, N ) for any β ∈ 0, α−1 α , and uα (∂Σ) ⊆ M . Furthermore, when k is sufficiently large, uαk is homotopic to uα , and hence (uα )∗ ∼ (uαk )∗ ∼ f∗ . On the other hand, from the weak compactness of the unit ball in W 1,2α (Σ, N ), a subsequence of {uαk } converges weakly to some uα in W 1,2α (Σ, N ). It follows that the two limits from the strong convergence and the weak convergence agree, that is uα = uα ∈ W 1,2α (Σ, N ). Thus uα is in Wα . Now from the lower semi-continuity of the α-energy, we have Eα (uα ) = Iα . 3.2.2 Euler-Lagrange equations for Eα Now we investigate the convergence of {uα } for a sequence α → 1. The key to obtaining a limiting harmonic map is to get estimates for a sequence of critical maps of Eα that are independent of α. For this, the coefficients in the Euler-Lagrange equations for Eα for a sequence α → 1 have to be uniformly controlled. When N is compact, these coefficients can be given in terms of the second fundamental form of some isometric embedding N → RK (cf. page 154, Colding-Minicozzi [12]). In the situation where the ambient manifold is noncompact, we shall employ an intrinsic version of the EulerLagrange equation for Eα given in terms of intrinsic geometric quantities of the ambient manifold. 21 3.2. Existence of minimizing harmonic maps in a conjugacy class We now give the intrinsic version of the Euler-Lagrange equations (cf. Fraser [22]). Let (x1 , x2 ) be the standard Euclidean coordinates on a disk D ⊂ Σ. Given p ∈ M , we choose Fermi coordinates f 1 , · · · , f n on an open neighbourhood V of p in N that satisfy the following: (i) V ∩ M is the zero set of f m+1 , · · · , f n (ii) the metric g on N in these coordinates satisfies gab (p) = 0 for any p ∈ M when a = 1, · · · , m and b = m + 1, · · · , n where gab = g( ∂f∂ a , ∂f∂ b )). Let Γabc denote the Christoffel symbols of N in a local coordinates. The Euler-Lagrange equation is given by 2 n ∆ua + Γabc i=1 b,c=1 α−1 ∂ub ∂uc + i i ∂x ∂x 1 + |du|2 2 i=1 a ∂ 2 ∂u (|du| ) =0 ∂xi ∂xi at interior points in Σ. If a disk is centred at a boundary point in ∂Σ ⊂ M , using Fermi coordinates, one can write the boundary conditions as ∂ua ∂r (x) = 0 ua (x) = 0 (B) for a = 1, · · · , m for a = m + 1, · · · , n. Proposition 3.2.2. uα ∈ C ∞ (Σ, N ). Proof. The result follows from Sacks and Uhlenbeck [48] for interior regularity, and from Fraser [22] for boundary regularity. Now we cover Σ by small disks of radius R in the interior, and half disks of radius R along the boundary, where locally we choose Fermi coordinates. When we scale these disks and half disks to unit size the energy integral becomes Eα (u) = R2(1−α) D (R2 + |∇u|2 )α dµ where D is the unit disk. The Euler-Lagrange equations appear in the following forms after this conformal dilation: 2 n ∆ua + Γabc i=1 b,c=1 ∂ub ∂uc α−1 + i i ∂x ∂x R + |du|2 2 i=1 a ∂ 2 ∂u (|du| ) =0 ∂xi ∂xi (3.2.1) or 2 a ∆u n ∂ub Γabc i + i=1 b,c=1 + 2gbc (u) ∂uc α−1 + i ∂x ∂x R + |du|2 ∂ 2 ub ∂uc ∂ua ∂xi ∂xj ∂xj ∂xi 2 n i,j=1 b,c=1 ∂(gbc ◦ u) ∂ub ∂uc ∂xi ∂xj ∂xj =0 22 3.2. Existence of minimizing harmonic maps in a conjugacy class with the boundary condition (B) on half disks in Fermi coordinates. Since N is allowed to be noncompact, we impose suitable conditions on N . A complete Riemannian manifold N is homogeneously regular if its injectivity radius is bounded from below and its sectional curvature is bounded (see [43] p. 623, [45]). Definition 3.2.3. A Riemannian manifold N is said to be homogeneously regular if there exist positive numbers λ, Λ and K, independent of p0 , such that each point p0 of N lies in an open set Bp0 in N which can be mapped onto the unit ball B1 (0) by a bi-Lipschitz map such that p0 corresponds to the origin and the following hold 1) λ|x|2 ≤ x, x 2) sup B1 (0) ∂gij , ∂xk 2 g(p) ≤ Λ|x| , for all ∂ 2 gij ≤ K, for i, ∂xk ∂xn p ∈ B1 (0), x ∈ Tp B1 (0); j, k, n = 1, 2, · · · , n, where g = gij dxi dxj is the metric on B1 (0) induced from that on Bp0 . With this condition, and the assumption that the boundary of the surface lies in a compact submanifold of N , we can derive the main estimate for critical maps of the α-energy at interior points of Σ in a similar manner as in the case of closed surfaces in compact manifolds (Proposition 3.2 of [48]). In fact, the smaller the disk, the nearer to Euclidean is the metric on the expanded disk. Thus a priori estimates are uniform in α ≥ 1 and 0 < R ≤ 1. In the boundary situation, let Dr+ = Σ ∩ Dr for a point x ∈ ∂Σ, where Dr the disk of radius r about x. If the total energy of u is small enough, its derivative is bounded on Dr+ by some constant C depending on r and the geometry of N in u(Dr+ ), by Lemma 1.6 in [22]. Since N is homogeneously regular, C = C(r, K, λ, Λ). By choosing r sufficiently small, we can assume u(∂Σ ∩ Dr ) is contained in a coordinate chart in N with Fermi coordinates defined as above. Therefore the Euler-Lagrange equations (3.2.1) with the boundary condition (B) are valid and we can derive the main estimate at boundary points (Proposition 1.7, Fraser [22]). 23 3.2. Existence of minimizing harmonic maps in a conjugacy class 3.2.3 Main estimates We state the main interior and boundary estimates (cf. Sacks-Uhlenbeck [48], Fraser [22]). Theorem 3.2.4. There exist > 0 and α0 > 1 such that: if uα : Σ → N with uα (∂Σ) ⊂ M is a critical map of Eα on Wα and E(u|D ) < , then there is an estimate uniform in 1 ≤ α ≤ α0 , ∇u ≤ K(p, D , D, N ) ∇u D ,1,p D,0,2 for any D ⊂ D , 1 < p < ∞. Proof. First we prove interior estimate, i.e. Σ ∩ D = ∅. We denote 2 n Γabc i=1 b,c=1 2 n i,j=1 b,c=1 ∂ub ∂uc = A(∇u, ∇u) ∂xi ∂xi ∂(gbc ◦ u) ∂ub ∂uc ∂ua ˙ = G(∇u, ∇u, ∇u) ∂xi ∂xj ∂xj ∂xi and 2 n 2gbc (u) i,j=1 b,c=1 ∂ 2 ub ∂uc ∂ua = G(∇2 u(∇u), ∇u) = G(∇2 u, ∇u, ∇u) ∂xi ∂xj ∂xj ∂xi ˙ and G are linear in each variable respectively, and they are where A, G, given in terms of the metric and its derivatives. Then we can write the Euler-Lagrange equation (3.2.1) as ∆u + A(∇u, ∇u) + α−1 ˙ G(∇u, ∇u, ∇u) + G(∇2 u, ∇u, ∇u) = 0 R + |∇u|2 (3.2.2) Denote by (D1 , D2 , ϕ) two disks D2 D1 and a smooth function ϕ on Σ, such that supp(ϕ) ⊂ D1 and ϕ = 1 on D2 . We compute ϕ∇u = ∇(uϕ) − u∇ϕ ϕ∆u = ∆(uϕ) − u∆ϕ − 2 ∇u, ∇ϕ ϕ∇2 u = ∇2 (uϕ) − u∇2 ϕ − 2∇u ⊗ ∇ϕ. 24 3.2. Existence of minimizing harmonic maps in a conjugacy class Multiplying (3.2.2) by ϕ, and inserting the above equalities, we get ∆(uϕ) − u∆ϕ − 2 ∇u, ∇ϕ + A ∇(uϕ) − u∇ϕ, ∇u α−1 + G˙ ∇(uϕ) − u∇ϕ, ∇u, ∇u R + |∇u|2 + G ∇2 (uϕ) − u∇2 ϕ − 2∇u ⊗ ∇ϕ, ∇u, ∇u = 0. Since |∇u|2 < R + |∇u|2 , we have ∆(uϕ) + A(∇uϕ, ∇u) α−1 + G˙ ∇uϕ, ∇u, ∇u + G ∇2 (uϕ), ∇u, ∇u R + |∇u|2 = u∆ϕ + 2 ∇u, ∇ϕ + A u∇ϕ, ∇u α−1 + G˙ u∇ϕ, ∇u, ∇u + G u∇2 ϕ + 2∇u ⊗ ∇ϕ, ∇u, ∇u R + |∇u|2 ˙ G) |u| + |∇u| + |u||∇u| ≤ K(ϕ, A, G, ˙ G and ϕ. From this we derive where K is a constant depending on A, G, ∆(uϕ) 0,p ≤ K (α−1) uϕ 2,p + |∇(uϕ)||∇u| 0,p + u 1,p + |u||∇u| 0,p . By the Lp estimates for second-order elliptic equations, (see for example Theorem 8.2 in Agmon [3], or Theorem 9.13 in Gilbarg-Trudinger [23]), the operator ∆−1 : W 0,p (D) → (W 2,p (D) ∩ W01,p (D)) is bounded, that is h 2,p ≤C ∆h 0,p + h 0,p for any h ∈ W 2,p (D) ∩ W01,p (D), where C is a constant depending on p, the elliptic constant λ for the Laplacian on D and the constant K in the theorem in [3]. We note that K (Definition 5.1 in [3]) involves bounds on the metric and its first and second order derivatives. Therefore C depends on p, the metric of N and its derivatives up to order two on the domain D (see also Gilbarg-Trudinger [23]). Therefore when α − 1 > 0 is sufficiently small, we get an estimate uϕ 2,p ≤K |∇(uϕ)||∇u| 0,p + u 1,p + |u||∇u| 0,p (3.2.3) where K is a constant depending on the metric of N and its derivatives up to order two on the domain D, ϕ and p. 25 3.2. Existence of minimizing harmonic maps in a conjugacy class Since we can assume the total energy of u is small enough, its derivative is bounded on Dr by some constant C depending on N and r, by Lemma 1.6 in [22]. But by assumption N is homogeneously regular. Therefore the derivative of u is bounded on Σ and we get uϕ 2,p |∇(uϕ)||∇u| ≤K 0,p + u . 1,p From the Holder inequality, we have |∇(uϕ)||∇u| for any λ, µ > 0 such that uϕ 2,p 1 λ 0,p 1 µ + ≤K ≤ ∇(uϕ) ∇u 0,λp 0,µp = 1. Thus ∇(uϕ) 0,λp ∇u 0,µp + u 1,p . (3.2.4) Now letting p = 2 − δ, λ = and using that uϕ uϕ 1, 2 2 and µ = , δ 2−δ is uniformly bounded by uϕ 1,2(2−δ)/δ 2(2−δ) δ ≤K Then for u such that ∇u uϕ 1, where 0 < δ < 2 ∇(uϕ) 0,2 K 2(2−δ) δ 0, ∇u 2(2−δ) δ 0,2 + u 2,2−δ , 1,2−δ we get . < 1, we get ≤ K 1 − ∇u u 0,2 K 1,2−δ . Or equivalently, for any p > 0 and u of small energy, we have an estimate uϕ 1,p ≤K u (3.2.5) 2p 1, p+2 where K depends on p, ϕ and the energy of u. Now let D be the unit disk on which the rescaled map u is defined, and D D D be two smaller disks. Applying the above argument to (D , D , ϕ) we get from (3.2.5) uϕ D ,1,p ≤ K1 u 2p D ,1, p+2 and applying the above argument to (D, D , φ) we get u D ,1,p = uφ D ,1,p ≤ uφ 1,p ≤ K2 u 2p 1, p+2 26 3.2. Existence of minimizing harmonic maps in a conjugacy class where we suppress the subscript of domain for D. This implies that u D ,1,µp ≤ K1 u uϕ D ,1,λp ≤ K2 u (3.2.6) 2µp 1, µp+2 . 2λp D ,1, λp+2 (3.2.7) From these two inequalities and (3.2.4) for (D , D , ϕ), we get u D ,2,p ≤ uϕ D ,2,p ≤ K ∇(uϕ) ≤ K u ≤ K u D ,0,λp u 2λp D ,1, λp+2 2λp 1, λp+2 u ∇u D ,0,µp 2µp 1, µp+2 2µp 1, µp+2 + u + u + u D ,1,p 2p 1, p+2 2p 1, p+2 where in the third inequality we have used (3.2.6) and (3.2.7) for the three norms. The Holder inequality implies f for 1 λ + 1 µ g 2λp λp+2 2µp µp+2 ≤ f 1 2 2 g 1 p. 1 2 2 1 p = 1, and f 2p p+2 ≤ f 2 Therefore we get u D ,2,p ≤ K ≤ K + u ≤ K u u 2λp 1, λp+2 u 2λp λp+2 2p p+2 u 2µp 1, µp+2 + ∇u + ∇u + u 2λp λp+2 2p 1, p+2 u 2µp µp+2 + ∇u 2µp µp+2 2p p+2 1,2 for any critical map of Eα with α − 1 > 0 and the energy of u sufficiently small. We can also write this estimate in an intrinsic form ∇u D ,1,p ≤ K ∇u 0,2 by considering an isometric embedding N → RK , choosing the origin such that D u = 0 and the Poincar´e inequality. 27 3.2. Existence of minimizing harmonic maps in a conjugacy class In the boundary situation, we consider two disks Dr and Dr around a boundary point with r < r, and a smooth function ϕ with support in Dr and is 1 on Dr , satisfying the boundary condition ∂ϕ ∂r = 0 on ∂Σ ∩ Dr . If the total energy of u is small enough, its derivative is bounded on Σ ∩ Dr by Lemma 1.6 in [22]. Then by choosing r sufficiently small, we can assume u(∂Σ ∩ Dr ) is contained in a coordinate chart in N with Fermi coordinates defined as above. Therefore the estimate (3.2.2) is available. Then we have (3.2.3) and by a similar argument, we have ∇u Dr ,1,p ≤ K ∇u Dr ,0,2 . 28 3.2. Existence of minimizing harmonic maps in a conjugacy class 3.2.4 Convergence of critical maps for Eα as α → 1 Next we consider the convergence of a sequence of critical maps of Eα as α → 1. Notice that for a sequence of minimizing maps uα of Eα we have a uniform energy bound. Let f0 be a smooth map in the homotopy class of f , which exists since C ∞ (Σ, N ) is dense in C(Σ, N ). Then f0 ∈ Wα for all α > 0. Since uα minimizes Eα on Wα , we have 1 + |∇uα |2 α 1 + |∇f0 |2 dµ ≤ Σ α dµ. Σ Then for α ∈ (1, 2), we get that the energy of uα is uniformly bounded as |∇uα |2 dµ ≤ Σ 1 + |∇uα |2 Σ α 1 + |∇f0 |2 dµ ≤ 2 dµ. Σ Lemma 3.2.5. Let uα be a sequence of critical maps of Eα with E(uα ) ≤ B. Then a subsequence uα → u strongly in L2 (Σ, RK ), weakly in W 1,2 (Σ, RK ), and E(u) ≤ lim E(uα ). α→1 We then have the following convergence result for critical maps of small energy. Lemma 3.2.6 ([48], [22]). Let uα : Σ → N with uα (∂Σ) ⊂ M be critical maps of Eα on Wα for a sequence α → 1, that converge weakly in L2 (Σ, RK ). Then there exists > 0 such that if E(uα |D ) < , then {uα } → u in C 1 (D r2 , N ) and u : D r2 → N is a smooth harmonic map such that u(D r2 ) meets M orthogonally along u(∂Σ ∩ D r2 ). We can now deduce global convergence of critical maps of the α-energy away from a finite number of points. Theorem 3.2.7 ([48], [22]). Let uα : Σ → N with uα (∂Σ) ⊂ M be critical maps of Eα for a sequence α → 1, that converge weakly in L2 (Σ, RK ), with E(uα ) < B. Then there exists a finite set of points {z1 , · · · , zl } of Σ such that uα → u in C 1 (Σ − {z1 , · · · , zl }, N ) and u : (Σ, ∂Σ) → (N, M ) is a smooth harmonic map satisfying the free boundary condition. Furthermore, when Σ is not a disk, if each uα induces the same action on the fundamental group as f , then so does u. Proof. The convergence part is Theorem 4.4 in [48] and Theorem 1.15 in [22]. We need only verify that the induced map of uα on the fundamental group is preserved in the limiting process, regardless of a finite set of points where bubbling may occur. 29 3.2. Existence of minimizing harmonic maps in a conjugacy class Choose, as generators of π1 (Σ), k + 2g loops through a base point ∗ in Σ such that all of these curves {γj } stay away from the points {z1 , · · · , zl } where the C 1 convergence fails. Since uα → u in C 1 (Σ − {z1 , · · · , zl }, N ), uα (∪j γj ) is homotopic to u(∪j γj ) for α sufficiently close to 1. It follows that there exists a path connecting uα (∗) and u(∗) such that uα (γj ) can be deformed to u(γj ) along the same path for each j. Therefore by definition, u induces the same action as uα , and thus f on π1 (Σ). On the other hand, π1 (Σ, ∂Σ) is the set of free homotopy classes of paths from a fixed point ∗ ∈ ∂Σ to a varying point on ∂Σ, and for each class a representative can be chosen away from the points {z1 , · · · , zl }. Therefore u induces the same action as uα for α sufficiently close to 1, and thus f on π1 (Σ, ∂Σ). Geometrically the condition of preserving the action on π1 is equivalent to the condition that the free homotopy class of Σ1 is preserved, where Σ1 is the union of a set of base-pointed generators for π1 . Since the 1-skeleton Σ1 can deform around any nontrivial S 2 in the ambient manifold without affecting the homotopy class of Σ1 , the action on π1 is not affected by π2 (N ) (whereas the homotopy class of Σ2 = Σ is changed). Therefore in the special case that minimizing sequences {uαk } for Eα , and minimizing sequences {uα } for E (but each need not be minimizing for Eα ) are considered in the admissible spaces Wfα and Wf with a fixed conjugacy class, we have the following result of C 1 -convergence. Corollary 3.2.8. Let uα be a sequence of minimizing maps of Eα in Wfα for a sequence α → 1, that converges in L2 (Σ, RK ). Then a subsequence uα → u in C 1 (Σ, N ) where u is a smooth harmonic map in Wf . Proof. Let f0 ∈ Wf be a smooth map. Since uα is minimizing, from the above we have E(uα ) ≤ E2 (f0 ) provided α ≤ 2. Thus by Theorem 3.2.7 there exists a subsequence uα → u in C 1 (Σ − {z1 , · · · , zl }, N ). Choose small disks or half disks Di (ρ) of radius ρ around each zi . Denote A = ∪Di (ρ) and B = Σ − A. We can get modified maps u ˜α that agree with uα on B 1 and such that u ˜α → u in C (Σ, N ) (for the details see the proof of Theorem 3.2.9). Choose a set of generators for π1 (Σ) and any nontrivial element of π1 (Σ, ∂Σ) away from these disks. Then (˜ uα )∗ ∼ (uα )∗ . Since uα is minimizing in Wfα , it follows that E(uα |Σ ) ≤ E(˜ uα |Σ ). Since u ˜α = uα on B, this implies E(uα |A ) ≤ E(˜ uα |A ). 30 3.2. Existence of minimizing harmonic maps in a conjugacy class On the other hand, by the C 1 -convergence we have E(˜ uα |A ) → E(u|A ) ≤ |A| max |∇u|2 . Hence for α − 1 sufficiently small E(uα |A ) ≤ (1 + σ)|A| max |∇u|2 ≤ (1 + σ)πρ2 l u 2 1,∞ where σ > 0 is a constant. Now we can choose ρ sufficiently small such that E(uα |A ) < , where is as in Lemma 3.2.6. Then by Lemma 3.2.6, we get uα → u in C 1 (Σ, N ). Remark. In comparison, blowup points for a sequence of minimizing maps in a fixed homotopy class can occur by the existence of nontrivial elements in π2 (N ) or π2 (N, M ). Therefore when Σ is a disk D, to show the action on π1 is preserved in the limit, the argument above cannot be applied as a blowup point may be on the generator of π1 (∂D), for a sequence of minimizing maps in a fixed homotopy class of the boundary curve. See Section 3.5. 3.2.5 Existence of a harmonic map in any homotopy class The convergence may fail at a finite number of interior or boundary points, where bubbling occurs. Thus the homotopy class of {uα } can be altered in the limiting process. However, under stronger conditions on the topology of N and M , we obtain the existence of a harmonic map in any homotopy class. This can be interpreted as the free boundary analog of Theorem 5.1 in [48]. Theorem 3.2.9. Let N , M and Σ be as in Theorem 1.0.2. If in addition π2 (N ) = 0 and π2 (N, M ) = 0, then there exists a minimizing harmonic map satisfying the free boundary condition in every free homotopy class of maps in C 0 ((Σ, ∂Σ), (N, M )). I. Analytic proof First we give a proof by modifying each uα in its homotopy class using the topological condition, to get a sequence of modified maps that converges in C 1 (Σ). Then we show the C 1 -convergence of {uα } by comparing the two sequences and the minimizing property of uα . Proof. Let uα be a minimizing map of Eα in a fixed homotopy class of f for a sequence α → 1. Then {uα } has uniformly bounded energy by the note after Proposition 3.2.1. By Theorem 3.2.7, there exists a finite set of points 31 3.2. Existence of minimizing harmonic maps in a conjugacy class {z1 , · · · , zl } such that a subsequence uα → u in C 1 (Σ − {z1 , · · · , zl }, N ) and u : (Σ, ∂Σ) → (N, M ) is a smooth harmonic map satisfying the free boundary condition. We claim that under the topological assumptions of the theorem, uα → u in C 1 (Σ, N ). At each point zi where the C 1 convergence fails, center a small disk Dρ ¯ ρ for j = i. in Σ about zi of radius ρ, where ρ is small enough so that zj ∈ /D First, assume zi is an interior point, and choose ρ small enough so that Dρ ∩ ∂Σ = ∅. Let η(r) be a smooth function that is 1 for r ≥ 1 and 0 for r ≤ 21 , and as in [48] Theorem 5.1, define a modified map uα by uα (z) = expu(z) η |z|/ρ exp−1 u(z) (uα (z)) , (3.2.8) where exp is the exponential map on N . Then uα agrees with uα outside Dρ and with u on Dρ/2 , and uα → u in C 1 (Dρ , N ). By assumption, π2 (N ) = 0 and so uα and uα are homotopic. If zi is a boundary point, let A ⊂ ∂Σ be the segments of the intersection of ∂Σ with the annulus { ρ2 < |z| < ρ}. For the map defined in (3.2.8), uα (A) may not lie in M , however we can modify the map so that it does ¯ ρ − Dρ/2 and satisfy the boundary condition. Since uα → u in C 1 on D ¯ u(∂Σ) ⊂ M , we may choose a neighborhood Ωα of A in Dρ − Dρ/2 that lies in a tubular neighborhood of ∂Σ and so that the nearest point projection from ∂Ωα ∩ int(Σ) to ∂Σ is one-to-one, such that uα (Ωα ) lies in a tubular neighborhood of M in N , with |Ωα | → 0 as α → 1. On Ωα , we redefine uα to map each geodesic segment between a point z of ∂Ωα − A and its nearest point in ∂Σ proportionally to the geodesic segment in N between uα (z) and its nearest point in M . This modified uα is piecewise smooth since u, uα , the exponential map, and the nearest point projection maps are smooth. ¯ ρ − Dρ/2 , |∇uα | is bounded independent Moreover, since uα → u in C 1 on D of α on Ωα , and since |Ωα | → 0 we have limα→1 Eα (uα |Ωα ) = 0. Finally, since uα = uα on the half circle {|z| = ρ}, the assumption π2 (N, M ) = 0 implies that there exists a homotopy between the disk uα (Dρ ) ∪ uα (Dρ ) and a disk in M , relative to the boundary uα (Dρ ∩ ∂Σ) ∪ uα (Dρ ∩ ∂Σ). Hence the two disks bound a 3-dimensional disk in N and there exists a homotopy between uα and uα relative to the half circle {|z| = ρ}, mapping the boundary Dρ ∩ ∂Σ into M . Therefore in either case, whether zi is an interior or boundary point of Σ, we have defined a map uα homotopic to uα such that ˜α (uα |Dρ ) = E(u|Dρ ), lim E α→1 32 3.2. Existence of minimizing harmonic maps in a conjugacy class ˜α (u) = where E (1 + |∇u|2 )α − 1 dµ. Since uα is minimizing for Eα in its homotopy class, Eα (uα |Dρ ) ≤ Eα (uα |Dρ ). Therefore, ˜α (uα |Dρ ) ≤ lim E ˜α (uα |Dρ ) = E(u|Dρ ) lim sup E(uα |Dρ ) ≤ lim sup E α→1 α→1 α→1 ≤ πρ2 u 2 1,∞ . Choose ρ sufficiently small so that πρ2 u 21,∞ < /2, where is as in Lemma 3.2.6. Then for α sufficiently close to 1, we have E(uα |Dρ ) < and by Lemma 3.2.6, uα → u in C 1 on Dρ . Hence uα → u in C 1 (Σ, N ) and u is in the same free homotopy class as f . II. Topological proof Lemma 3.2.10. Suppose N is a manifold, M is a submanifold of N , and Σ is a compact orientable surface. Let f0 , f1 : Σ → N be two maps which induce the same action on the fundamental group π1 (Σ) → π1 (N ). (i) If π2 (N ) = 0, then f0 and f1 are homotopic. (ii) Suppose ∂Σ = ∅, and f0 , f1 map ∂Σ to M . If furthermore a homotopy {ft }t∈[0,1] exists such that each boundary component Cs contains a point qs , for which the curve {ft (qs )}t∈[0,1] is null-homotopic in π1 (N, M ), and we assume π2 (N, M ) = 0, then a homotopy between f0 and f1 can be chosen such that ft (∂Σ) ⊂ M, ∀t ∈ [0, 1]. Proof. We note that the Lemma is trivial when Σ is the sphere or the disk. Thus we consider compact orientable surfaces which are not a sphere or a disk. (i). We deal with (g ≥ 1, k = 0), (g = 0, k ≥ 2) and (g ≥ 1, k ≥ 1) respectively. The fundamental polygon for a compact surface with g ≥ 1 and k = 0 is a 4g-sided domain, with the edges labelled as −1 −1 σ1 , σ2 , σ1−1 , σ2−1 , · · · , σ2g−1 , σ2g , σ2g−1 , σ2g where {σi }1≤i≤2g is a set of generators for π1 (Σ). Then the condition that f0 and f1 induce the same action (see the definition in section 3.2) implies that f0 ( i σi ) is homotopic to f1 ( i σi ), i.e. there exists a homotopy φ: σi × I → N, φ(x, 0) = f0 |∪σi (x), φ(x, 1) = f1 |∪σi (x). 33 3.2. Existence of minimizing harmonic maps in a conjugacy class Then we can extend this map to a polyhedron σi ∪ (Σ, 0) ∪ (Σ, 1) → N φ: by letting φ(x, 0) = f0 (x), φ(x, 1) = f1 (x) for x ∈ Σ. Moreover, by the fact that a polyhedron is homeomorphic to S 2 and the assumption π2 (N ) = 0, we can extend φ to the solid polyhedron φ˜ : Σ × I → N, ˜ 0) = f0 (x), φ(x, ˜ 1) = f1 (x). φ(x, It follows that f0 and f1 are homotopic. Surfaces of g = 0 and k ≥ 2 can be realized as a k-sided polygon with the edges labelled as σ1 , · · · , σk , where {σi }1≤i≤k is a set of generators for π1 (Σ) satisfying the relation i σi = 1 (so it is a free group with k − 1 generators). By a similar argument, we can show f0 and f1 are homotopic. Suppose g ≥ 1 and k ≥ 1 (Σ has non-empty boundary). Then Σ can be realized as a 4g-sided polygon with k non-intersecting disks {Dj } in the interior of this polygon removed. We denote by Sj the loops formed by attaching some segments from the boundary of these disks to the vertex of the polygon, i.e. the base point. Then the edges σi and the loops Sj form a set of generators for π1 (Σ). Since f0 and f1 induce the same action, there exists a homotopy between f0 and f1 on A = i σi ∪ j Sj g k σi ∪ φ: i=1 Sj × I → N, φ(x, 0) = f0 |A (x), φ(x, 1) = f1 |A (x). j=1 Then by a similar argument, f0 and f1 are homotopic, as the curve A can be considered as the boundary of a disk-type surface. (ii). We consider the boundary problem (Σ, ∂Σ) → (N, M ) assuming ∂Σ = ∅. The hypothesis in the theorem asserts the existence of a homotopy ϕ : Σ × I → N, ϕ(x, 0) = f0 (x), ϕ(x, 1) = f1 (x). (3.2.9) By assumption, each boundary circle Cs contains a point qs ∈ Cs such that ϕ({qs } × I) is homotopic to a segment λs ⊂ M connecting f0 (qs ) and f1 (qs ), by a homotopy relative to the end points ϕs : I × I → N, ϕ (t, 0) = ϕ(qs , t), ϕ (t, 1) = λs (t) where the first factor I refers to the domain for the curve, and the second factor I refers to a parameter of the deformation. Next we shall modify the homotopy ϕ in (3.2.9) such that {qs } × I is mapped to λs ⊂ M . 34 3.2. Existence of minimizing harmonic maps in a conjugacy class We denote the the the the the half disk D+ diameter S half circle C half ball B + base disk D0 {(x, y, z) ∈ B : x = 0, z ≥ 0} {(x, y, z) ∈ B : x = 0, z = 0} {(x, y, z) ∈ S 2 : x = 0, z ≥ 0} {(x, y, z) ∈ B : z ≥ 0} {(x, y, z) ∈ B : z = 0} where B is the unit ball in R3 . Then S = D+ ∩ D0 and C = D+ ∩ S 2 . Now we extend ϕ to (Σ × I) ∪ B + in steps. a). Since ϕs is relative to the end points, the domain for this homotopy can be considered as D+ such that S represents the domain for the curve ϕ({qs } × I) and C represents the domain for the curve λs ⊂ M . Identifying S ⊂ D0 with {qs } × I ⊂ ∂Σ × I, we may consider D0 as a subset of ∂Σ × I. Since ϕ is defined on Σ × I, we have that ϕ|D0 and ϕs |D+ agree on S = D0 ∩ D+ and this defines a map ϕ|D0 ∪ ϕs |D+ , or ϕ|Σ×I ∪ ϕs |D+ . b). We take B + as the union of two balls B1+ = {(x, y, z) ∈ B + : x ≤ 0}, B2+ = {(x, y, z) ∈ B + : x ≥ 0} and let S1 = D+ ∪ B1+ ∩ D0 , S2 = D+ ∪ B2+ ∩ D0 be two half spheres of B1+ and B2+ respectively. Then S1 ∪ S2 = D0 ∪ D+ . Moreover, there exist retraction maps Π1 : B1+ → S1 , Π2 : B2+ → S2 such that the pull-back of the map ϕ|D0 ∪ ϕs |D+ on Si by Πi defines a map ϕs : B + → N which agrees with ϕ on D0 = (Σ × I) ∩ B + . Therefore the union of the two maps ϕ|Σ×I ∪ ϕs |B + defines a map. c). We still denote this union map ϕ|Σ×I ∪ ϕs |B + by ϕ. Since there is a retraction from B + to D0 , the adding of B + doesn’t affect the topological type of Σ × I, and so the domain for ϕ can be considered as Σ × I. More importantly, since D0 is in ∂Σ × I, ϕ0 is again a homotopy between f0 and f1 , as ϕ(·, t) is kept for t = 0, 1. But then {qs } × I will be mapped to λs ⊂ M . Now the loop f0 (Cs ) · λs · f1 (Cs ) · λ−1 s ⊂M 35 3.2. Existence of minimizing harmonic maps in a conjugacy class forms the boundary of a disk ϕ : Cs × I → N. Recall that the condition π2 (N, M ) = 0 implies any disk D1 with boundary in M is homotopic to a point in M by a homotopy from D1 × I to N . Identifying D1 × {1} to a point, we get a map from a 3-dimensional cone to N with base disk D1 such that ∂D × {t} is mapped to M for any t ∈ [0, 1]. Thus the union of the boundary circles is a disk D2 ⊂ M , which has the same boundary ∂D × {0} as D1 . Since the two disks bound a ball, there exists a homotopy between them relative to the boundary. Using this fact and that the disk ϕ(Cs × I) lies entirely in M , we get a relative homotopy ϕ˜s : {Cs × I} × I → N satisfying the initial condition ϕ˜s (p, 0) = ϕ|Cs ×I (p), p ∈ Cs × I (3.2.10) ϕ˜s (p, 1) ∈ M, p ∈ Cs × I (3.2.11) ϕ˜s (x, 0), t = f0 (x), x ∈ Cs , t ∈ I (3.2.12) ϕ˜s (x, 1), t = f1 (x), x ∈ Cs , t ∈ I (3.2.13) and the boundary condition as the boundary loop is (Cs × {0}) ∪ (Cs × {1}) ∪ ({qs } × I). Now by (3.2.10), we can consider the union of ϕ|Σ×I and ϕ˜s |Cs ×I , s = 1, · · · , k, as a homotopy defined on Σ × I, since there exists a retraction from {Cs × I} × I to Cs × I. By (3.2.12) and (3.2.13) this is a homotopy {ft }0≤t≤1 between f0 and f1 , and from (3.2.11) satisfying ft (∂M ) ⊂ M for all t ∈ [0, 1]. Proof of Theorem 3.2.9. Let uα be as in the proof in Subsection 3.2.5 such that uα is a minimizing map for Eα in a given homotopy class f and uα → u in C 1 (Σ − {z1 , · · · , zq }, N ). Then uα and u induce the same action by Theorem 3.2.7 and satisfy the hypotheses in Lemma 3.2.10 for α − 1 sufficiently small, where the initial homotopy as in (ii) can be defined by a linear homotopy in the tangent bundle through the exponential map (where the condition π2 (N ) = 0 is used). Then we can choose a point p in every boundary circle of ∂Σ such that the geodesic connecting uα (p) and u(p) lies in a tubular neighborhood of M for α − 1 sufficiently small. Then by Lemma 3.2.10 there exists a homotopy between u and uα in the space C 0 ((Σ, ∂Σ); (N, M )), and u is in the prescribed homotopy class f . 36 3.2. Existence of minimizing harmonic maps in a conjugacy class Remark 1). The requirement that the homotopy in (ii) take boundary value in M is because we minimize the energy in the space C((Σ, ∂Σ); (N, M )). Remark 2). From the exact homotopy sequence: π ∂ i · · · → π2 (M ) → π2 (N ) → π2 (N, M ) → π1 (M ) → π1 (N ) → · · · we can deduce that if π2 (N, M ) = 0, then i : π1 (M ) → π1 (N ) is injective. In particular, there is no minimizing disk with free boundary in M . 3.2.6 Convergence of harmonic maps for varying conformal structures We will need the following convergence result for harmonic maps with respect to varying conformal structures on Σ. Theorem 3.2.11. Let ui : (Σ, ∂Σ) → (N, M ) be a harmonic map satisfying the free boundary condition for a conformal structure ci on Σ, where Σ is not of disk type. Suppose ci converges to a conformal structure c in the C ∞ -topology and E(ui , ci ) ≤ B. Then there exist a subsequence {ui } and a finite set of points {z1 , · · · , zl } such that ui → u in C 1 (Σ − {z1 , · · · , zl }, N ), where u : (Σ, ∂Σ) → (N, M ) is a smooth harmonic map satisfying the free boundary condition, and E(u, c) ≤ lim inf i→∞ E(ui , ci ). Furthermore, if each ui induces the same action on the fundamental group as f , then so does u. Proof. The convergence for varying conformal structures on a closed surface lying in a bounded set is shown in [49] Theorem 2.3, and the argument carries through for bordered surfaces. The argument that the induced map of ui on the fundamental group is preserved in the limiting process is as in the proof of Theorem 3.2.7 above. As in Corollary 3.2.8, we also have a stronger result for a sequence of minimizing maps. Corollary 3.2.12. Let ci be a sequence of conformal structures and ui be a minimizing map for ci in Wf . Suppose ci converges to a conformal structure c. Then a subsequence ui → u in C 1 (Σ, N ) where u is a smooth harmonic map. Proof. Let f0 ∈ Wf be a smooth map. Since ui is minimizing for ci , ci → c and E(h, ·) is continuous, we get E(ui , ci ) ≤ E(f0 , ci ) → E(f0 , c), and so E(ui , ci ) is uniformly bounded. Since ci → c, a subsequence {ui } converges 37 3.2. Existence of minimizing harmonic maps in a conjugacy class weakly in L2 (Σ, RK ) with respect to the conformal structure c. From Theorem 3.2.11, ui → u in C 1 (Σ − {z1 , · · · , zl }, N ). Choose small disks or half disks Di (ρ) of radius ρ around each zi . Denote A = ∪Di (ρ) and B = Σ − A. We can define modified maps u ˜i that agree with ui on B and such that 1 u ˜i → u in C (Σ, N ) (see the proof of Theorem 3.2.9). We can choose a set of generators for π1 (Σ) and any nontrivial element in π1 (Σ, ∂Σ) away from these disks. Then (˜ ui )∗ ∼ (ui )∗ . Since ui is minimizing in Wf , it follows that E(ui |Σ , ci ) ≤ E(˜ ui |Σ , ci ). Since u ˜i = ui on B, this implies E(ui |A , ci ) ≤ E(˜ ui |A , ci ). Let φi : (A, ci ) → (A, c) be a conformal map with respect to the pullback conformal structure φ∗i (c) = ci on each disk or half disk. Since ci → c, 1 we have φi → id. Then u ˜i ◦ φ−1 i → u in C (A, N ) and so 2 E(˜ ui ◦ φ−1 i |A , c) → E(u|A , c) ≤ |A| max |∇u| . By conformal invariance, we have ∗ E(ui ◦ φ−1 i |A , c) = E(ui |A , φi (c)) = E(ui |A , ci ) and E(˜ ui ◦ φ−1 ui |A , φ∗i c) = E(˜ ui |A , ci ). i |A , c) = E(˜ Combining the above, we have for sufficiently large i E(ui ◦ φ−1 i |A , c) = E(ui |A , ci ) ≤ E(˜ ui |A , ci ) = E(˜ ui ◦ φ−1 i |A , c) ≤ (1 + σ)|A| max |∇u|2 ≤ (1 + σ)πρ2 l u 2 1,∞ where σ > 0 is a constant. Now we can choose ρ sufficiently small such that E(uα |A ) < , where is as in Lemma 3.2.6. Then since φi → id, for sufficiently large i, we have E(ui |A ) < and by Lemma 3.2.6, ui converges to u in C 1 (A, N ). 38 3.3. A variational problem on the space of conformal structures M for compact Riemann surfaces 3.3 3.3.1 A variational problem for conformal structures on compact Riemann surfaces The notion of Riemann moduli space and Teichm¨ uller space A conformal structure on a smooth surface is a collection of local charts {fα : Uα → C} such that the transition maps are holomorphic. Let M0 be an orientable surface. A conformal structure on M0 can also be considered as an element (M, f ), that is f M0 −−−−→ M where f is a diffeomorphism and M is a Riemann surface with a conformal structure c. In this sense the two factors (M, c) and f determine a conformal structure on M0 (to make it a Riemann surface). But if we ignore the diffeomorphism f , we get an element (M, c). The space of all such elements is called the Riemann moduli space for M0 . With the pull-back conformal structure, the diffeomorphism f (M0 , f ∗ c) −−−−→ (M, c) f f is holomorphic. Suppose M0 −−− −→ M and M0 −−−−→ M represent the same conformal structure, that is we have (M0 , f ∗ c) f −−−−→ (M, c) id f (M0 , (f )∗ c ) −−−−→ (M , c ) −1 Then M −f−◦f −−→ M is biholomorphic. Therefore (M, c) and (M , c ) represent the same element if and only if there exists a biholomorphic map between them. Suppose two conformal structures on M0 are given by equivalent elements (M, c) and (M , c ). In the above sense, we don’t distinguish (M, c) and (M , c ) and will write (M, c). Then we will have a commutative diagram f M0 −−−−→ M id h f ◦h M0 −−−−→ M 39 3.3. A variational problem on the space of conformal structures M for compact Riemann surfaces where h is a diffeomorphism of M0 . Writing c = f ∗ c, then h : (M0 , h∗ c ) → (M0 , c ) is holomorphic. Therefore we can also say two conformal structures c and c on M0 are equivalent if and only if there exists a diffeomorphism h of M0 such that h : (M0 , c ) → (M0 , c) is holomorphic. Now we define the Riemann moduli space of a Riemann surface. Let Σ be an orientable surface. Let D(Σ) denote the topological group of diffeomorphisms of Σ onto itself with the C ∞ -topology of uniform convergence on compact sets of all differentials. Let M(Σ) denote the space of conformal structures on Σ. There is a natural action M(Σ) × D(Σ) → M(Σ) by pulling back conformal structures. The Riemann moduli space of Σ is defined as the quotient R(Σ) = M(Σ)/D(Σ) consisting of equivalent conformal structures with respect to this action. With the natural topology, the Riemann moduli space is not a manifold for any Riemann surface of Euler characteristic χ(Σ) ≤ 0. The Teichm¨ uller space is the universal covering space of any conceivable moduli space of a Riemann surface, and has the structure of a manifold. Precisely, let D0 (Σ) denote the subgroup of D(Σ) consisting of orientationpreserving diffeomorphisms which are homotopic to the identity. Then the Teichm¨ uller space is defined as T (Σ) = M(Σ)/D0 (Σ). The Teichm¨ uller space and the Riemann moduli space are related by R = T /Mod where Mod = D/D0 is the Teichm¨ uller modular group. An equivalent definition of the Teichm¨ uller space can be formulated by reducing the homotopy condition to the conjugacy condition. A marking on a Riemann surface Σ is a generating set of simple closed curves for π1 (Σ). We denote by Σ1 the union of these generators. Two markings are equivalent if and only if the 1-skeletons Σ1 for the markings are freely homotopic. We say two marked Riemann surface (Σ, c, Σ1 ) and (Σ, c , Σ1 ) are equivalent if and only if there exists a holomorphic diffeomorphism f : (Σ, c) → (Σ, c ) such that f (Σ1 ) and Σ1 are equivalent. Then one can show that the equivalence classes of marked surfaces form a space which is diffeomorphic to the Teichm¨ uller space of Σ defined as above. 40 3.3. A variational problem on the space of conformal structures M for compact Riemann surfaces For bordered surfaces, there are two kinds of Teichm¨ uller spaces. We denote the following subgroups of D(Σ): D0 (Σ) : orientation-preserving diffeomorphisms that are homotopic to the identity and map each boundary curve onto itself; D0 (Σ; ∂Σ) : orientation-preserving diffeomorphisms that are homotopic to the identity and keep the boundary fixed; D0 (Σ; U ) : orientation-preserving diffeomorphisms that are homotopic to the identity and keep a subset U ∂Σ fixed. The unreduced Teichm¨ uller space is defined as T (Σ) = M(Σ)/D0 (Σ; ∂Σ) and the reduced Teichm¨ uller space is defined as T ∗ (Σ) = M(Σ)/D0 (Σ; U ) (= M(Σ)/D0 (Σ) for U = ∅). For example, when Σ is the annulus, M(Σ)/D0 (Σ; x0 ) is (0, ∞) where x0 is any point on ∂Σ, and M(Σ)/D0 (Σ; ∂Σ) is infinite-dimensional. The reduced Teichm¨ uller space and the Riemann moduli space are related by R(Σ) = ∗ T (Σ)/Mod(Σ) where Mod(Σ) is the reduced Teichm¨ uller modular group. Definition 3.3.1. Specifically for a compact bordered Riemann surface Σ of χ(Σ) ≤ 0, we define 1) T ∗ (Σ) = M(Σ)/D0 (Σ), 2) T ∗ (Σ) = M(Σ)/D0 (Σ; x0 ), if χ(Σ) < 0, if it is the annulus. The study of Teichm¨ uller spaces for bordered surfaces can be reduced to one without boundary. The construction is due to Schottky. Namely, a Riemann surface 2Σ without boundary can be formed as the union of a bordered Riemann surface Σ and an exact duplicate of it, with a conformal structure 2c induced by that on Σ. This conformal structure is symmetric in the sense that there is an antiholomorphic diffeomorphism S of 2Σ such that S 2 = id and it leaves ∂Σ fixed. Let MS be the space of conformal structures on 2Σ, for which S is antiholomorphic and D0S be the subgroup of D0 consisting of diffeomorphisms which commute with S. Then T ∗ ∼ = MS /D0S (see e.g. [55]). 3.3.2 The structure of M for compact bordered surfaces We have the following theorem concerning the structure of M(Σ) (cf. [16]). 41 3.3. A variational problem on the space of conformal structures M for compact Riemann surfaces Theorem 3.3.2 (Earle-Schatz). Suppose Σ is a compact bordered Riemann surface of Euler characteristic χ(Σ) ≤ 0. Then (i) M(Σ) is a contractible Fr´echet manifold, (ii) D0 (Σ) or D0 (Σ; x0 ) acts freely, continuously, and properly on M(Σ), (iii) M(Σ) → M(Σ)/D0 (Σ) = T ∗ (Σ) is a principal D0 (Σ) fibre-bundle. M(Σ) → M(Σ)/D0 (Σ; x0 ) is a principal D0 (Σ; x0 ) fibre-bundle. In both cases, it follows that M(Σ) is the universal D0 (Σ) (or D0 (Σ; x0 )) bundle. On the other hand, it is known that T ∗ (Σ) is homeomorphic to R6g−6+3k for χ(Σ) < 0, and R for the annulus. Then from the theory of fibre bundles, M(Σ) is homeomorphic to T ∗ (Σ) × D0 (Σ) (or T ∗ (Σ) × D0 (Σ; x0 )) and D0 (Σ) (or D0 (Σ; x0 )) is contractible. Corollary 3.3.3. Suppose Σ is a compact bordered Riemann surface. Then M(Σ) has a trivial bundle structure M(Σ) ∼ = T ∗ (Σ) × D0 (Σ) if χ(Σ) < 0, and M(Σ) ∼ = T ∗ (Σ) × D0 (Σ; x0 ) if Σ is the annulus. Remark 3.3.4. Parallel theorems for closed Riemann surfaces of g > 1 were proved in Earle-Eells [15]. Later Fischer-Tromba ([20]) showed the homeomorphisms in these theorems were indeed diffeomorphisms. The differential geometric version of a Teichm¨ uller theory for bordered Riemann surfaces of χ(Σ) < 0 was proved in Tomi-Tromba [55] using the Schottky construction (see also section 4.3, [14]). The structure theory of conformal structures implies the following result on the metrics (see [48] for the case of closed Riemann surfaces). Corollary 3.3.5. Suppose Σ is a compact bordered Riemann surface of χ(Σ) < 0. Let g(t) be a variation of a metric g = g(0) on Σ. Then every such variation arises from a composition of (i) the pull-back of metric by a C ∞ family of diffeomorphisms in D0 , (ii) a smooth curve in the Teichm¨ uller space, (iii) a family of conformal changes in the metric. Proof. Let P be the space of C ∞ positive functions and G be the space of smooth metrics on Σ. Any two metrics g and g are conformal if g = λg for some λ ∈ P and a representative metric in a conformal class {λg} can be chosen to have Gauss curvature as −1. These metrics form a space M−1 = G/P. From [55], M−1 is diffeomorphic to M. Therefore a variation 42 3.3. A variational problem on the space of conformal structures M for compact Riemann surfaces of metrics decomposes as a family of conformal changes and a smooth curve in M. But by the above results, the latter decomposes as a smooth curve in T ∗ and the pull-back by a C ∞ family of diffeomorphisms in D0 . 3.3.3 Variational problems on the space of conformal structures In some variational problems, one often needs to produce critical points or minimizers for certain functionals defined on the space of conformal structures M (possibly with some function space). This can be done by proving the convergence of a minimizing sequence of conformal structures for the functional. A natural idea is to work with the total space M. However, this space is infinite-dimensional and the compactification problem is rather difficult. Thus we often have to look at smaller spaces which allow for compactification, so that we can get a convergent minimizing sequence. For compact Riemann surfaces, there are two candidates for this role, the Riemann moduli space R and the (reduced) Teichm¨ uller space T ∗ , as both spaces allow for compactification. Then we have two different situations. i) Subspace. The first approach is to define the functional on R or T ∗ by considering the space as a “subspace” of M. Let π denote the projection π : M → M/D = R, π : M → M/D0 = T ∗ . Suppose we can “embed” R or T ∗ into M such that an equivalent class [x] is mapped to a conformal structure cx . A natural condition will be that cx ∈ [x]. Although this can be realized by choosing an arbitrary cx for each [c], we require that the map be continous (and smooth), since we should also consider the topology (and geometry), such as the compactification (and differentiable features of the manifold and the functional). Hence the proposition that R or T ∗ can be embedded as a subspace of M is equivalent to the existence of a continuous (smooth) section i : R → M, i:T∗→M such that π ◦ i = id. Here we also have to differ the cases of R and T ∗ . From Theorem 3.3.2, we know that D0 acts freely, continuously and properly on M, and hence M → M/D0 defines a principal D0 fibre-bundle. In comparison, the action of D is only effective ∗ and M → M/D doesn’t define a principal bundle. ∗ A group action G × X → X is free if no element in G has a fixed point in X, i.e. ∀(g, x) ∈ G × X, g · x = x; and it is effective if ∀g ∈ G, g = id. 43 3.3. A variational problem on the space of conformal structures M for compact Riemann surfaces For the Teichm¨ uller space, from the theory of fibre-bundles, we know that the existence of a continuous (smooth) section i : T ∗ → M such that π ◦ i = id is equivalent to the fact that the principal bundle M is homeomorphic (diffeomorphic) to the trivial bundle T ∗ × D0 (Corollary 8.3, [28]). On the other hand, this is true by Theorem 3.3.2 (and Remark 3.3.4). Thus T ∗ can be realized as a subspace (smooth submanifold) of M in a natural way so that it inherits the topology (geometry) of the total space and is homeomorphic (diffeomorphic) to R6g−6 . Therefore any functional on the space of conformal structures (and possibly some function space) can be defined on the Teichm¨ uller space as a subspace of the former space. ii) Quotient space. For any functional H on M (and possibly some function spaces X), to define it on R or T ∗ as a quotient space of M, the functional should be invariant with respect to the action of D or D0 . Namely, we should have H(x, f ∗ c) = H(x, c) for any x ∈ X and f ∈ D (or D0 ), which is a very strong condition. However, if we only care about obtaining a critical point or a minimizer for H on X × M, we can instead look at a modified functional ¯ H(c) = inf H(·, c), X c∈M ¯ ∗ c) = H(c) ¯ with the invariance condition that H(f for any f ∈ D (or D0 ). If ¯ this is satisfied, then H is well-defined on R (or T ∗ ). Then the strategy is to first obtain a convergent minimizing sequence ¯ in R (or T ∗ ) in the compactified space, and furthermore by addifor H tional assumptions prove that the limiting [c] is not on the boundary of the compactified space and is in R (or T ∗ ). Then some representative conformal structure c for [c] and a minimizer for H(·, c) will probably produce a minimizer for H on X × M. For example, consider the energy functional E : W 1,2 (Σ, N )×M(Σ) → R for a compact orientable surface Σ E(u, c) = |∇g u|2g dµg where g is a Riemannian metric∗ compatible with c and dµ is the area element induced by g. ∗ A conformal structure c on a surface induces an almost complex structure J, given by dφ−1 ◦ i ◦ dφ where {φ} is a chart for c and i is the multiplication by i on the complex plane. Then in dim-2 J determines uniquely a class of symmetric positive (2, 0)-tensors (metrics). By the conformal invariance of E, we can choose any such a metric. 44 3.3. A variational problem on the space of conformal structures M for compact Riemann surfaces ¯ ∗ c) = H(c) ¯ The condition H(f is now inf E(u, c) = inf E(u, f ∗ c). u∈X (3.3.1) u∈X Since E(u, f ∗ c) = E(u ◦ f −1 , c) by the conformal invariance of E, this is inf E(u, c) = inf E(u ◦ f −1 , c) = u∈X u∈X inf u∈Xf −1 E(u, c) where Xf = {x ◦ f : x ∈ X} for a diffeomorphism f ∈ D (or D0 ). We give two examples of X. a). X = W 1,2 (Σ, N ), where N is a Riemannian manifold. Then Xf = X, ∀f ∈ D and the condition (3.3.1) is satisfied. Thus E is well-defined on W 1,2 (Σ, N ) × M/D1 for any subgroup D1 of D. In particular, E can be defined on both W 1,2 (Σ, N ) × R and W 1,2 (Σ, N ) × T ∗ . b). X is the function space with a fixed conjugacy class of a given continuous map h X = {u ∈ W 1,2 (Σ, N ) | u(∂Σ) ⊆ M, u∗ ∼ h∗ } where M ⊂ N is a compact submanifold. Then we have Xf = {u ◦ f ∈ W 1,2 (Σ, N ) | u ◦ f (∂Σ) ⊆ M, (u ◦ f )∗ ∼ h∗ } = {u ∈ W 1,2 (f (Σ), N ) | u(f (∂Σ)) ⊆ M, u∗ ∼ h∗ ◦ f∗−1 } = X ∀f ∈ D0 (recall that D0 consists of diffeomorphisms of Σ that map each boundary curve onto itself and are homotopic to the identity). Thus the condition (3.3.1) is satisfied and E can be defined on W × T ∗ . In comparison, E cannot be defined on W × R as the boundary and the homotopy conditions are not satisfied for all f ∈ D. Now the spaces R and T ∗ as quotient spaces of M (as the definitions for them), both allow for compactification. Thus whenever the condition (3.3.1) is satisfied for some functional H, we can define H on R and T ∗ in a natural way. Note that in the subspace approach, the discussion doesn’t depend on the functional. 3.3.4 The minimal area problem, first approach: the functional E¯ on the space M Recall the Euler characteristic of a surface of genus g with k boundary components is χ(Σ) = 2−2g −k. On the disk D, there is only one conformal 45 3.3. A variational problem on the space of conformal structures M for compact Riemann surfaces structure, and any smooth harmonic map u : D → N with u(∂D) ⊂ M and meeting M orthogonally along u(∂D) is conformal, and hence a branched minimal immersion. When Σ is not a disk, that is when χ(Σ) ≤ 0, in order to produce a branched minimal immersion, we must vary the conformal structure on Σ. Let M(Σ) denote the space of conformal structures on Σ with the C ∞ topology as in the above. Given a conformal structure c ∈ M(Σ), let g be a Riemannian metric compatible with c and dµ be the area element induced by g. We then consider E : W 1,2 (Σ, N ) × M(Σ) → R where |∇g u|2g dµ. E(u, c) = Σ By virtue of the conformal invariance of the energy functional, this is independent of the choice of the metric g and is well-defined. We note that E(·, c) is lower semi-continuous in u, and E(u, ·) is continuous in c. For each conformal structure c we have produced, by Proposition 3.2.1 and Theorem 3.2.7, when Σ is not a disk, a smooth harmonic map uc in the admissible space Wf = {u ∈ W 1,2 (Σ, N ) | u(∂Σ) ⊆ M, u∗ ∼ f∗ } such that E(uc , c) = inf E(u, c) (note that for f ∈ W 1,2 (Σ, N ), f∗ can be u∈Wf defined as in section 1, Schoen-Yau [50]). Now we define a functional on the ¯ : M(Σ) → R by space of conformal structures E ¯ E(c) = inf E(u, c) = E(uc , c). u∈Wf (3.3.2) ¯ is continuous on M(Σ). Lemma 3.3.6. E ¯ i) = Proof. Let ci → c be a sequence of conformal structures. Suppose E(c ¯ E(ui , ci ) and E(c) = E(u, c) for some ui , u ∈ Wf . Let K = lim inf E(ui , ci ), and let {(uik , cik )} be a subsequence such that E(uik , cik ) → K. By Theorem 3.2.11 there exists a further subsequence, which we continue to denote by 46 3.3. A variational problem on the space of conformal structures M for compact Riemann surfaces {uik }, and u0 ∈ Wf such that E(u0 , c) ≤ limk→∞ E(uik , cik ). Then, E(u, c) = lim E(u, ci ) i ≥ lim sup E(ui , ci ) (since ui is minimizing for ci ) i ≥ lim inf E(ui , ci ) i = lim E(uik , cik ) k ≥ E(u0 , c) ≥ E(u, c) (since u is minimizing for c) ¯ ¯ i ). It follows that E(c) = E(u, c) = lim E(ui , ci ) = lim E(c i i ¯ is attained at c ∈ M(Σ). Let u be a minimizing harSuppose inf E M(Σ) monic map for c. Then for any pair (u , c ) ∈ Wf × M(Σ), we have ¯ ¯ ) = inf (·, c ) ≤ E(u , c ). E(u, c) = inf (·, c) = E(c) ≤ E(c Wf Wf The relationship between such a minimizing pair and the minimal immersion problem is illustrated in the following (cf. Theorem 1.8 [48] and Corollary 1.5 [49]). Theorem 3.3.7. If (u, c) is a critical point of E on Wf × M(Σ), then u is a branched minimal immersion. Proof. Any variation of the metric arises from a composition of a conformal change in the metric, and a curve in M(Σ). Hence by the conformal invariance of E, the fact that c is a critical point of E(u, ·) on M(Σ) implies that E is critical with respect to any variation of the initial metric induced by c. The computation of Sacks and Uhlenbeck ([48], p.6) shows that u is weakly conformal in the interior of Σ. Then by Gulliver-Osserman-Royden [24], u is a branched minimal immersion. Corollary 3.3.8. If (u, c) is a minimizer of E on Wf × M(Σ) with respect to all smooth variations of c preserving the action on the fundamental group, then u minimizes area among all branched immersions having the same action. ¯ ¯ i ) → inf E. ¯ If a subseLet {ci } be an E-minimizing sequence, i.e. E(c ¯ ¯ by Lemma quence converges to a conformal structure c, then E(c) = inf E 3.3.6. In fact it suffices to have a weaker condition that {ci } converges in the level of the moduli space. 47 3.3. A variational problem on the space of conformal structures M for compact Riemann surfaces ¯ Lemma 3.3.9. Let {ci } be an E-minimizing sequence. If there exist diffeomorphisms φi ∈ D(Σ) and c ∈ M(Σ), such that φ∗i ci → c in the C ∞ ¯ topology, then there exists a minimizing conformal structure for E. Proof. Let ui be a minimizing harmonic map for φ∗i ci which induces the same action as f ◦ φi . Then ui ◦ φ−1 induces the same action as f and we i have ¯ i ) ≤ E(ui ◦ φ−1 , ci ) = E(ui , φ∗i ci ) ≤ E(uc ◦ φi , φ∗i ci ) = E(uc , ci ) = E(c ¯ i) E(c i i i where uci denotes a minimizing harmonic map for ci which induces the same action as f . Therefore, ¯ i ). E(ui , φ∗i ci ) = E(c (3.3.3) By Theorem 3.2.11 and Corollary 3.2.12, there exists a subsequence uik such that uik → u in C 1 (Σ), and since φ∗i ci → c, we have E(u, c) ≤ lim inf E(uik , φ∗ik cik ). (3.3.4) −1 For sufficiently large k, u ◦ φ−1 ik induces the same action as uik ◦ φik on the fundamental group by the proof of Theorem 3.2.11, hence is in the admissible space Wf , since ui ◦ φ−1 induces the same action as f . Then we have i ¯ −1 )∗ c) ≤ E(u ◦ φ−1 , (φ−1 )∗ c) E((φ ik ik ik = E(u, c) ≤ lim inf E(uik , φ∗ik cik ) ¯ i ) = lim inf E(c k = ¯ inf E M(Σ) where the first equality is by conformal invariance of the energy, the second inequality is by (3.3.4), the second equality is by (3.3.3), and the third ¯ Therefore equality follows since {cik } is a minimizing sequence of E. ¯ −1 )∗ c) = inf E. ¯ E((φ ik M(Σ) This proves the existence of a minimizing conformal structure. In fact, for ∗ ¯ all large k, (φ−1 ik ) c are E-minimizers. Therefore by Theorem 3.3.7, Corollary 3.3.8 and Lemma 3.3.9, the minimal area problem is reduced to the convergence problem in the moduli space R(Σ). 48 3.3. A variational problem on the space of conformal structures M for compact Riemann surfaces 3.3.5 The minimal area problem, second approach: the functional E¯ on the space T ∗ ¯ In the first approach to the minimal area problem, we utilize the functional E defined on the total space of conformal structures M, and only consider the moduli space later for the convergence problem. However, there is another approach of defining the functional initially on a compactifiable space. From the discussion in subsection 3.3.3, we know that the admissible space Wf is invariant with respect to the action of the group D0 , and using ¯ on T ∗ by the conformal invariance of E, we can define E ¯ E([c]) = inf E(·, c), Wf ∀[c] ∈ T ∗ where c is any representative conformal structure for [c]. Now let [ci ] be a ¯ ¯ i ]) → inf T ∗ E ¯ as i → ∞. In the compactE-minimizing sequence, that is E([c ∗ ification space of T , there exists some point [c0 ] such that a subsequence [ci ] → [c0 ]. If we can prove [c0 ] is not on the boundary of the compactification space, it is in T ∗ . Let c0 be any representative conformal structure for ¯ 0 ) = inf T ∗ E. ¯ Then we can choose a minimizing harmonic [c0 ]. We have E(c map u with respect to c0 , and we can show that (u, c0 ) is a minimizing pair for E on the space Wf × M. Moreover, from the above discussion we know that the minimizing con¯ on M (see the definition (3.3.2)) are exactly the formal structures for E ¯ on T ∗ . fibres (action by D0 ) over the minimizers for E ¯ on T ∗ as a quotient space by This allows one to define the functional E D0 . But we can also adopt a different setup. We know E is well-defined on Wf × T ∗ by using the section i : T ∗ → M which exists by Theorem 3.3.2 and Remark 3.3.4 (see the discussion in subsection 3.3.3). Then we can define ¯ E(c) = inf E(·, c), ∀c ∈ T ∗ → M Wf by considering T ∗ as a subspace of M. In the compactification space of T ∗ , using a similar argument as above, we obtain a minimizing sequence ci → c0 in T ∗ . But this is i(ci ) = ci → c0 = i(c0 ) in M. Therefore by ¯ 0 ) = lim E(c ¯ i ) and c0 is a minimizing conformal structure Lemma 3.3.6, E(c ∗ ¯ on T . Now we use the conformal invariance of E to get that c0 is a for E ¯ on M. This argument relies crucially minimizing conformal structure for E on the structure theory of M. We now prove part (i) of Theorem 1.0.2. 49 3.4. Minimal surfaces of non-disk type 3.4 Minimal surfaces of non-disk type ¯ Throughout we let {ci } be an E-minimizing sequence of conformal structures, and ui will denote a minimizing map for ci , with E(ui , ci ) < B. Now ¯ for nonwe prove the existence of a minimizing conformal structure for E disk surfaces. The first proof is based on the theory of moduli space for closed Riemann surfaces via the Schottky construction. 3.4.1 First proof I. Σ is not a cylinder. Assume that Σ is a surface with χ(Σ) < 0. For each conformal structure ci in the minimizing sequence, consider the doubled conformal surface. Applying the compactification theorem of the moduli space of conformal structures for the closed doubled conformal surfaces (Lemma 4 of Abikoff [1]), there is a subsequence of {ci } (which we continue to denote by {ci }) and there are diffeomorphisms φi of Σ such that either φ∗i ci → c in C ∞ or, (Σ, φ∗ ci ) converges to a Riemann surface with nodes Σ∞ corresponding to pinching a set of homotopically nontrivial simple closed curves in the doubled surface to nodes wm , m = 1, . . . n. In the first case, by Lemma 3.3.9 we are done. In the second case, we have curves γm in Σ which are pinched, each of which is either a closed curve (possibly a boundary component) or a curve between two boundary components of Σ (corresponding to a closed curve in the doubled surface that crosses ∂Σ, and must be reflection invariant across ∂Σ). We may then argue as in [49] Theorem 4.3. We may choose a nested sequence {Djm } of closed neighborhoods of γm such that Djm converges to the node wm of Σ∞ , and for each fixed j, the change in the conformal structure on Σ as (Σ, φ∗i ci ) → Σ∞ is restricted to the interior of ∪nm=1 Djm ([5]). Let Σj = Σ − ∪nm=1 Djm . By Theorem 3.2.11 and Corollary 3.2.12, there is a (1) subsequence {ui } of {ui } that convergences in C 1 (Σ1 , N ) to a smooth har(j−1) monic map defined on Σ1 . Given {ui }, by Theorem 3.2.11 and Corollary (j) (j−1) 3.2.12, a subsequence {ui } of {ui } converges in C 1 (Σj , N ) to a smooth (i) harmonic map. Consider the diagonal sequence {ui } which converges to a 1 harmonic map u in C (Σ∞ , N ), where Σ∞ is the punctured Riemann surface Σ − {w1 , . . . , wn }. Since E(ui , ci ) < B for all i, E(u) < B, and by Theorem 1.6 of [48] and Theorem 1.10 of [22], u can be extended to a smooth ˜ ∞ → N satisfying the free boundary condition, where harmonic map u : Σ ˜ Σ∞ = Σ∞ ∪ {q1 , . . . , qs , (qs+1 , qs+1 ), . . . , (qn , qn )} is the bordered Riemann surface obtained by adding a point qm at the punctures of Σ∞ corresponding 50 3.4. Minimal surfaces of non-disk type to the nodes wm ∈ Σ∞ resulting from the pinching of components of ∂Σ, and adding a pair of points (qm , qm ) at the two punctures of Σ∞ (which may be boundary points) corresponding to each node wm ∈ Σ∞ resulting from the pinching of a closed curve inside Σ or a closed curve in the doubled surface that crosses ∂Σ. Now let γ be a curve homotopic to γm , for any fixed m between 1 and n, chosen to lie in Djm for j sufficiently large. Since ˜ ∞ is homotopically trivial (either as a closed curve or a relative curve γ⊂Σ between boundary components) it follows that u(γ) is homotopically trivial. (i) (i) But limi→∞ ui (γ) = u(γ), so ui (γ) is homotopically trivial for i sufficiently large. Since γ is homotopically nontrivial in Σ, this contradicts our assumption that the induced map on the fundamental groups is injective. Therefore the second case cannot occur. II. Σ is a cylinder. A cylinder with a conformal structure can be represented by a parallelogram spanned by the vectors (1, 0) and ξ in R2 with sides corresponding to one of the two generators identified. Two cylinders given by ξ1 , ξ2 , with the same corresponding sides identified, represent conformally equivalent cylinders if ξ2 = τ ξ1 for some τ ∈ PSL(2, Z). Given our minimizing sequence of conformal structures ξi , and associated minimizing harmonic maps ui , there exist elements τi ∈ PSL(2, Z) such that τi ξi lies in the fundamental domain of PSL(2, Z). If Im(τi ξi ) ≤ b < ∞ for all i, then a subsequence of {τi ξi } converges to η, and by Lemma 3.3.9 we are done. Otherwise, suppose that κi = Im(τi ξi ) → ∞. Let ηi = τi ξi . Then vi = ui ◦ τi−1 : (Σ, ηi ) → N is harmonic and E(vi , ηi ) = E(ui , ξi ) ≤ B. We consider the following two cases: a) If the sides corresponding to ηi are identified, then on any cylinder S 1 × [0, κ) we can find a subsequence of {vi } which converges in C 1 (S 1 × [0, κ), N ) to a harmonic map v : S 1 × [0, κ) → N with E(v) < B. Since κ was arbitrary, using a diagonal sequence argument as above, we obtain a harmonic map v : S 1 × [0, ∞) → N with E(v) < B. But S 1 × [0, ∞) ¯ − {p} for some p ∈ D, and by Theorem 1.6 in [48], v is conformally D extends to a smooth harmonic map v : D → N providing a homotopy of vi (S 1 × {q}) v(S 1 × {q}) to a point for suitable q and i sufficiently large. This implies that the generator τi−1 (S 1 ×{q}) of (Σ, ξi ) is mapped by ui , and hence also by f , to a loop homotopic to zero, contradicting the assumption that f∗ : π1 (Σ) → π1 (N ) is injective. b) If the sides corresponding to (0, 1) are identified, then on any strip 51 3.4. Minimal surfaces of non-disk type [0, 1] × (−κ, κ) we can find a subsequence {vi } which converges in C 1 ([0, 1] × (−κ, κ), N ) to a harmonic map v : [0, 1] × (−κ, κ) → N ) with E(v) < B. Since κ was arbitrary, we can obtain a harmonic map v : [0, 1] × R → N ¯ − {p1 , p2 } for some p1 , p2 ∈ with E(v) < B. But [0, 1] × R is conformally D ∂D, and by Theorem 1.10 in [22], v extends to a smooth harmonic map ¯ → N providing a homotopy of vi ([0, 1] × {q}) v :D v([0, 1] × {q}) to a point for suitable q and i sufficiently large. This contradicts the assumption that f∗ : π1 (Σ, ∂Σ) → π1 (N, M ) is injective. 3.4.2 Second proof In the second proof, we look at the moduli space for a bordered surface that is not a cylinder, in the setup of hyperbolic geometry, and the argument is more geometric. I. Σ is not a cylinder. Recall that for each conformal structure ci on Σ, we can form the doubled surface (2Σ, 2ci ) (cf. Abikoff [2]). We will use this construction later. By the existence of a hyperbolic metric on any compact Riemann surface with χ < 0, we can assume a hyperbolic metric gi on Σ, induced from ci . Then the following facts hold: i) Every closed geodesic or geodesic connecting two boundary curves, with respect to this hyperbolic metric, has a collar (a component of a tubular neighborhood minus the geodesic) of area ≥ k1 > 0; ii) Any curve that is homotopically nontrivial in π1 (N ) or π1 (N, M ) has length ≥ k2 > 0; iii) E(ui , ci ) ≤ B. Specifically the first fact is from Keen’s theorem (page 98 [2], also [33]) and by considering a geodesic connecting two boundary curves as exactly half of a closed geodesic in the doubled surface, the second is implied by the homogeneously regular condition on N , and the third is since E(ui , ci ) → ˜ These facts mark the ingredients of a computation by Schoen and inf E. Yau (Lemma 3.1, [50]) on the length of closed geodesics in Σ, which also applies to geodesics connecting two boundary curves. To be precise, since f∗ : π1 (Σ) × π1 (Σ, ∂Σ) → π1 (N ) × π1 (N, M ) is injective, any curve in a collar as in i) that is homotopically nontrivial in π1 (Σ) or π1 (Σ, ∂Σ), is mapped to a curve in N that is homotopically 52 3.4. Minimal surfaces of non-disk type nontrivial in π1 (N ) or π1 (N, M ) respectively. Hence by ii), the length of this curve in N is bounded from below. Then it follows from i) and iii) (cf. Lemma 3.1, [50]) that there is a positive lower bound on the length of any closed geodesic in Σ or geodesic connecting two boundary curves of Σ. In particular, this holds for all boundary curves of Σ, which are geodesics with respect to the hyperbolic metric. Note that any closed geodesic in the doubled surface (2Σ, 2ci ) of (Σ, ci ) arises from a closed geodesic in (Σ, ci ) or the doubling of a geodesic connecting two boundary curves of (Σ, ci ). It follows that there is a positive lower bound on the length of any closed geodesic in the doubled surface. Then by Mumford’s compactness theorem ([46], and page 271, [14] for a differential geometric description and a boundary discussion), there exist a subsequence {gi } and diffeomorphisms φi of Σ onto itself, such that φ∗i gi converges in C ∞ to a hyperbolic metric g on Σ. Let c be the conformal structure associated with g. Since the space of hyperbolic metrics and the space of conformal structures are diffeomorphic for compact Riemann surfaces with χ(Σ) < 0, the sequence {φ∗i ci } converges to c with the C ∞ -topology. II. Σ is a cylinder. It is known that the moduli space of a cylinder is (0, 1) and the conformal structures can be realized as {Ar }0<r<1 , where Ar = {z| r ≤ |z| ≤ 1} is an annulus. Suppose ci is a conformal structure Ari (in the moduli space). Then {ri } has a limit in [0, 1], which is either 0, or 1, or r ∈ (0, 1). a). If a subsequence ri → r ∈ (0, 1), then ci → Ar in the moduli space. Hence by Lemma 3.3.9, there exists a minimizing conformal structure. b). If a subsequence ri → 0, then by Theorem 3.2.7 and Corollary 3.2.8, there exists a subsequence {u1,i } that converges to a harmonic map in Ar1 . Given {uk−1,i }, a subsequence {uk,i } converges in Ar2 . Then a diagonal subsequence {ui,i } converges to a smooth harmonic map u in C 1 (D − {0}). But by the removable singularity theorem of Sacks and Uhlenbeck ([48]), since E(u) ≤ B, u extends to D. It follows that C = {|z| = 1} will be mapped to a homotopically trivial curve in π1 (N ), and for sufficiently large i, ui (C) is homotopically trivial in π1 (N ). On the other hand, the induced map of ui is injective on π1 (Σ) × π1 (Σ, ∂Σ, ∗), which implies that C can’t be mapped to a curve which is homotopically trivial in π1 (N ). Thus we get a contradiction. c). If a subsequence ri → 1, then we get a sequence of maps ui : Ari → N with E(ui , Ari ) ≤ B. Denote by Cθ the segment arg z = θ in Ari , and lθ the 53 3.5. Minimizing disks length of ui (Cθ ) in N . Suppose N → Rk is an isometric embedding. Then, 2π 2π (lθ )2 dθ = 0 0 Cθ ∂ui dr ∂r 2π ≤ Cθ Cθ ∂ui 2 dr dθ ∂r 2π = (1 − ri ) 0 ≤ = 1 − ri 2π ri 0 Cθ 1 − ri · E(ui ). ri dθ ∂ui 2 dr dθ ∂r (1 − ri ) 0 2 ∂ui ∂θ 2 1 ∂ui + 2 r ∂r 2 rdrdθ By the assumption E(ui ) ≤ B, the measure of the set of θ for which lθ > λ > 0 goes to zero as ri → 1, for any λ > 0. Hence for sufficiently large i, there exists θ such that ui (Cθ ) lies entirely in a geodesic ball centered at one end point, and thus can be deformed to a path in M relative to the two end points. Therefore ui (Cθ ) is homotopically trivial in π1 (N, M ). Recall that π1 (Σ, ∂Σ, ∗) is the set of free homotopy classes of paths in Σ, from the base point ∗ ∈ ∂Σ to a varying point in ∂Σ. When Σ is the cylinder, it consists of two elements, i.e. the trivial map ∗ and any segment connecting the two boundary curves, such as Cθ . Then the injectivity on π1 (Σ, ∂Σ, ∗) implies that Cθ can’t be mapped to a path which is homotopically trivial in π1 (N, M, ∗). Thus we get a contradiction. Hence {ci } has a limit in the moduli space. We have proved part (i) of Theorem 1.0.2. 3.5 Minimizing disks Now we prove part (ii) of Theorem 1.1, that there exists a set of free homotopy classes {Γj } of closed curves in M such that the elements {γ ∈ Γj } form a generating set for ker i∗ , where i∗ : π1 (M ) → π1 (N ) is the homomorphism induced by the inclusion, and each Γj can be represented by the boundary of an area minimizing disk that solves the free boundary problem (D, ∂D) → (N, M ). We will need the following lemma. 54 3.5. Minimizing disks 3.5.1 An integral identity for critical maps of Eα Lemma 3.5.1. Let u : (D, ∂D) → (N, M ) be a critical map of Eα on W 1,2α (D, ∂D; N, M ). Then u satisfies − 1 + |∇u|2 α + α 1 + |∇u|2 α−1 |∇u|2 z dxdy = 0 (3.5.1) D where z = x + iy is the complex coordinate on the disk D. Proof. Writing u = u(z, z¯), we have |∇u|2 = |ux |2 + |uy |2 = 4uz · uz¯ and 1 + 4uz · uz¯ Eα (u) = i dzd¯ z. 2 α D Given a complex number β, let ϕβ (z) = z−β ¯ . 1 − βz Let β(t) be a differentiable curve in C with |β(t)| < 1, β(0) = 0. Then ϕt = ϕβ(t) is a family of automorphisms of the unit disk, which map the boundary to the boundary. Now we define a variation of u by ϕt u D −→ D −→ N z → w = ϕt (z) → u(w, w) ¯ where z = ϕ−1 t (w) = Then we have ∂w 1 − |β|2 = ¯ 2, ∂z (1 − βz) w+β ¯ . 1 + βw ∂z 1 − |β|2 = ¯ 2, ∂w (1 + βw) and using β(0) = 0, we compute ∂ ∂w ∂t ∂z = 2β¯ (0)z, t=0 ∂ ∂z ∂t ∂w = −2β¯ (0)w. (3.5.2) t=0 We have: Eα (u ◦ ϕt ) = 1+ D ∂w ∂ w ¯ |∇u|2 ∂z ∂ z¯ α ∂z ∂ z¯ i dwdw. ¯ ∂w ∂ w ¯ 2 55 3.5. Minimizing disks We compute ¯ 2 (1 + β w) ¯ ∂w ∂ w (1 + βw) ¯ 2 , = ∂z ∂ z¯ (1 − |β|2 )2 ∂ ∂t ∂w ∂ w ¯ ∂z ∂ z¯ = 2β¯ (0)w + 2β (0)w. ¯ t=0 Using this and (3.5.2), we have d Eα (u ◦ ϕt ) dt t=0 1 + |∇u|2 = α D i − 2β¯ (0)w − 2β (0)w ¯ dwdw ¯ 2 α|∇u|2 1 + |∇u|2 + α−1 2β¯ (0)w + 2β (0)w ¯ D 2 − 1 + |∇u|2 = β (0) α + α 1 + |∇u|2 α−1 i dwdw ¯ 2 |∇u|2 z¯ dxdy D +β¯ (0) 2 − 1 + |∇u|2 α + α 1 + |∇u|2 α−1 |∇u|2 z dxdy. D Since β (0) is arbitrary, we get (3.5.1). 3.5.2 Nontrivial limit of critical maps of Eα Now we use the integral identity in Lemma 3.5.1 to show that the limiting map of a sequence of nontrivial critical maps for Eα , in C 1 on the disk minus a boundary point, is not a constant map. This result will be crucial in proving that any free homotopy class in the kernel of i : π1 (M ) → π1 (N ) can be decomposed as free homotopy classes represented by minimizing disks. Corollary 3.5.2. Let uα be a sequence of critical maps of Eα for a sequence ¯ − {p}, N ) where p ∈ ∂D, and each uα is α → 1. If uα → u in C 1 (D nontrivial, then u is not a constant map. Proof. From (3.5.1), taking the imaginary part, and using the fact that D y dxdy = 0, we have − 1 + |∇uα |2 α + 1 + α 1 + |∇uα |2 α−1 |∇uα |2 y dxdy = 0. D Note that the integrand is similar to that in the variation formula for the sphere derived in Sacks-Uhlenbeck (Lemma 5.3, page 20, [48]). Thus by the same argument, we have for 1 ≤ α ≤ 2, α − 1 + |∇uα |2 + 1 + α 1 + |∇uα |2 α |∇uα |4 ≤ α−2 2 (α − 1) 1 + |∇uα |2 α−1 |∇uα |2 ≤ |∇uα |4 . 56 3.5. Minimizing disks Without loss of generality, we can assume p is the point (0, 1) ∈ ∂D ⊂ R2 . Dividing D into the upper half disk D+ and the lower half disk D− , we have α 2 1 + |∇uα |2 α−2 |∇uα |4 y dxdy ≤ − 1 + |∇uα |2 α−2 |∇uα |4 y dxdy. D− D+ Assume u is a constant map. Then uα cannot converge to u in C 1 (D, N ) (Theorem 1.8 in [22]). Therefore p is a blowup point: that is (by Lemma 1.16 and p. 957 in [22]), bα = max |∇uα (z)| = |∇uα (zα )| → ∞ z∈D where limα→1 zα = p. Consider the rescaled maps u ˜α (z) = uα (zα + b−1 α z). As α → 1, the domains of u ˜α exhaust either the whole plane or a half plane, and (a subsequence) {˜ uα } converges in C 1 on compact subsets to a nontrivial harmonic map u ˜. If DR (0) denotes the disk of radius R centered 57 3.5. Minimizing disks at the origin in the plane or the half plane, then we have 1 1 E(˜ u|DR (0) ) ≤ lim α→1 2 2 |∇˜ uα |2 dxdy DR (0) 1 = lim α→1 2 |∇uα |2 dxdy DR/bα (zα ) = lim α→1 1 2 |∇uα |2 − |∇uα |2 dxdy 1 + |∇uα |2 DR/bα (zα ) = lim α→1 |∇uα |2 · |∇uα |2 dxdy 1 + |∇uα |2 1 2 DR/bα (zα ) α α→1 2 ≤ lim 1 + |∇uα |2 α−1 1 + |∇uα |2 α−2 |∇uα |2 · |∇uα |2 y dxdy 1 + |∇uα |2 DR/bα (zα ) = lim α→1 α 2 |∇uα |4 y dxdy DR/bα (zα ) α α→1 2 ≤ lim 1 + |∇uα |2 α−2 1 + |∇uα |2 α−2 |∇uα |4 y dxdy D+ ≤ lim − α→1 |∇uα |4 y dxdy D− =0 where the first equality is by the conformal invariance of the energy functional, the second equality follows since bα → ∞, and the last equality follows since u is a constant map and uα → u in C 1 on D− , so |∇uα |2 → 0 uniformly. This contradicts the fact that u ˜ is nontrivial. Therefore u is not a constant map. 3.5.3 Decomposition of the kernel of i∗ : π1 (M ) → π1 (N ) by minimizing disks Now we come back to the specific setting of minimizing disks. Given a basepoint x0 ∈ M , let i∗ : π1 (M, x0 ) → π1 (N, x0 ) 58 3.5. Minimizing disks be the homomorphism induced by the inclusion i of M in N . Recall that two elements γ and γ in π1 (M, x0 ) determine the same free homotopy class of closed curves in M if and only if they belong to the same orbit π1 (M, x0 )γ = π1 (M, x0 )γ under the usual action of π1 (M, x0 ) on π1 (M, x0 ). That is, the set of free homotopy classes of closed curves in M is in oneto-one correspondence with the set of orbits π1 (M, x0 )γ ⊂ π1 (M, x0 ) (for further details see [48] p.19). Given an element γ in ker i∗ , let Γ be its associated free homotopy class. Let WΓ = {u ∈ W 1,∞ (D, ∂D; N, M ) : [u(∂D)] = Γ}, where we use the notation [u(∂D)] for the free homotopy class of u(∂D), and define ˜α (u) : u ∈ WΓ } E(Γ) = min {E(u) : u ∈ WΓ } = lim min {E α→1 ˜α (u) = ((1 + |∇u|2 )α − 1)dµ. Note that E(Γ) = 0 if and only if Γ where E is trivial, and E(Γ) > 0 otherwise ([22] Theorem 1.8). Lemma 3.5.3. Let γ ∈ ker i∗ and let Γ = π1 (M, x0 )γ be its associated free homotopy class. Then either Γ can be represented by the boundary of an area minimizing disk solving the free boundary problem, or for any δ > 0 there exist nontrivial free homotopy classes Γ1 = π1 (M, x0 )γ1 , Γ2 = π1 (M, x0 )γ1 where γ1 , γ2 ∈ ker i∗ , such that π1 (M, x0 )γ ⊂ π1 (M, x0 )γ1 + π1 (M, x0 )γ2 , E(Γ1 ) + E(Γ2 ) < E(Γ) + δ. Proof. Since γ ∈ ker i∗ there exists f : (D, ∂D) → (N, M ) such that [f (∂D)] = Γ. As in the proof of Proposition 3.2.1, there exists a minimizing map uα of Eα on WΓ . By Theorem 3.2.7 there exists a sequence α → 1 such that uα → u in C 1 on D minus a finite set of points, and u : (D, ∂D) → (N, M ) is a (possibly trivial) harmonic map satisfying the free boundary condition. If the set of points where the convergence fails is empty, then u is nontrivial, and hence is an area minimizing disk solving the free boundary problem with [u(∂D)] = Γ. Otherwise there exists a point p at which uα fails to converge to u in C 1 . Note that p cannot be an interior point. If p is an interior point, then as in the proof of Theorem 3.2.9 we can define a modified map uα by (3.2.8) with uα |∂D = uα |∂D , so uα ∈ WΓ and by the same argument as in the the proof of Theorem 3.2.9 we have uα → u in C 1 (Dρ (p), N ). Therefore, p ∈ ∂D. Now observe that given ρ > 0, we can find a neighborhood B of p in ¯ with |B| < ρ, that contains no other points where the convergence fails, D, 59 3.5. Minimizing disks ¯ → B leaving and such that there is a conformal diffeomorphism h : D − B ∂B ∩ D fixed. The existence of B and h can be seen in the following way. Let ϕ : D → H be a conformal map from the open disk D to the upper half plane H such that p is mapped to the origin and two nearby points q and q ∈ ∂D on either side of p are mapped to 1 and −1. We may choose q and ¯ has area q sufficiently close to p so that B := ϕ−1 (D+ ), where D+ = D ∩ H, less than ρ and contains no other points where the convergence fails. Let ¯ + → H − D+ be the conformal map S(z) = 1/¯ S:D z , which is the identity map on the half circle. Then we may take h = ϕ−1 ◦ S ◦ ϕ. Using the construction from Theorem 3.2.9, we can define a map uα that agrees with uα outside B and with u on neighborhood of p in B, and so that ˜α (uα |B ) = E(u|B ). Now define limα→1 E u1α = uα uα on D − B on B u2α = uα ◦ h uα on D − B on B. Let Γ1 and Γ2 be the free homotopy classes of u1α (∂D) and u2α (∂D) respectively. Then Γ ⊂ Γ1 + Γ2 . By the conformality of h, we have ˜α (u1 ) = lim E ˜α (uα |D−B ) + E(u|B ) lim E α α→1 α→1 ˜α (u2 ) = lim E ˜α (uα |B ) + E(u|B ). lim E α α→1 α→1 Choose ρ sufficiently small so that E(u|B ) ≤ u Then if α is sufficiently close to 1, we have 2 |B| 1,∞ < u 2 ρ 1,∞ < δ/6. ˜α (u1α ) ≤ E ˜α (uα |D−B ) + δ E 3 δ 2 ˜α (uα ) ≤ E ˜α (uα |B ) + , E 3 and ˜α (u1α ) + E ˜α (u2α ) ≤ E ˜α (uα ) + 2δ < E(Γ) + δ, E(Γ1 ) + E(Γ2 ) ≤ E 3 (3.5.3) where the last inequality follows since {uα } is a minimizing sequence for ˜α (u2 ) ≥ E(Γ). We may assume δ < 12 min{ , 0 }. By Lemma 3.2.6, E α ˜α (uα |B ) ≥ E(uα |B ) ≥ for α close to 1, and so E ˜α (u1 ) ≤ E(Γ) + δ − < E(Γ). E(Γ1 ) ≤ E α 60 3.5. Minimizing disks Therefore Γ1 = Γ and Γ2 is nontrivial. It remains to show that Γ1 is nontrivial. For α sufficiently close to 1, we have ˜α (u1 ) ≥ E ˜α (uα |D−B ) ≥ E(u|D−B ) − δ > E(u) − δ . E α 6 3 If u is nontrivial then E(u) ≥ 0 ([22] Theorem 1.8), and so ˜α (u1 ) > E(u) − δ ≥ E α 3 0 − δ > δ. 3 If u is trivial, by Corollary 3.5.2 there must be a second point p = p where the convergence uα → u fails, and p ∈ ∂D − B. Then by Lemma 3.2.6, ˜α (u1 ) ≥ E ˜α (uα |D−B ) ≥ E(uα |D−B ) ≥ for α close to 1. In either case, we E α 1 ˜ have Eα (uα ) > δ, and then by equation (3.5.3), ˜α (u2 ) < E(Γ) + δ − E(u ˜ 1 ) < E(Γ) + δ − δ = E(Γ). E(Γ2 ) ≤ E α α Therefore, Γ2 = Γ and Γ1 is nontrivial. Theorem 3.5.4. There exists a set of free homotopy classes {Γj } of closed curves in M such that the elements {γ ∈ Γj } form a generating set for ker i∗ acted on by π1 (M, x0 ), and each Γj can be represented by the boundary of an area minimizing disk that solves the free boundary problem. Proof. Let {Γj } be the free homotopy classes that can be represented by the boundary of an area minimizing disk that solves the free boundary problem. Let P ⊂ ker i∗ be the subgroup generated by γ ∈ Γj . Suppose P is a proper subgroup. Let I = inf E(Γ) over all free homotopy classes Γ with elements γ ∈ Γ, γ ∈ / P . Then there exists Γ such that E(Γ) < I + 0 /2. By assumption, Γ cannot be represented by the boundary of an area minimizing disk that solves the free boundary problem, and so by Lemma 3.5.3 there exist nontrivial Γ1 and Γ2 with π1 (M, x0 )γ ⊂ π1 (M, x0 )γ1 + π1 (M, x0 )γ2 and E(Γ1 )+E(Γ2 ) < E(Γ)+ 0 /2. Since Γ1 and Γ2 are nontrivial, E(Γj ) ≥ 0 for j = 1, 2. This implies E(Γj ) < E(Γ) − 0 /2 < I. Therefore, by assumption the sets π1 (M, x0 )γj are both in P , and so π1 (M, x0 )γ ⊂ π1 (M, x0 )γ1 + π1 (M, x0 )γ2 ⊂ P, a contradiction. Therefore P = ker i∗ , and so the elements {γ ∈ Γj } form a generating set for ker i∗ acted on by π1 (M, x0 ), such that each Γj can be represented by the boundary of an area minimizing disk that solves the free boundary problem. 61 3.6. Topology of minimal surfaces of low index 3.6 Topology of minimal surfaces of low index Let N be a compact 3-manifold with smooth boundary ∂N . Suppose Σ is a compact orientable two-sided minimal surface in N with boundary ∂Σ in ∂N solving the free boundary problem (Σ, ∂Σ) → (N, ∂N ). We will investigate controlling the genus and the number of boundary components of Σ for stable and index 1 minimal surfaces, under certain curvature and boundary assumptions on N . Let A denote the second fundamental form, and ν denote the unit normal vector field of Σ in N . Let η denote the outward unit conormal of Σ and T the unit tangent vector along ∂Σ. The index form is the quadratic form |∇f |2 − Ric(ν) + |A|2 f 2 dµ + I(f, f ) = Σ ∇ν ν, η f 2 ds ∂Σ for any normal variational vector field f ν. The index of Σ is defined as the number of negative eigenvalues of the associated bilinear form. 3.6.1 The first eigenfunction of the index form A function f ∈ W 1,2 (Σ, R) is an eigenfunction of the index form with eigenvalue λ if I(f, g) = λ f, g L2 for all g ∈ W 1,2 (Σ, R), that is f is a weak solution to the following equation ∇f · ∇g − (Ric(ν) + |A|2 )f g dµ + Σ ∇ν ν, η f g ds = λ ∂Σ f g dµ. Σ Since all coefficients are smooth up to the boundary, it follows that any eigenfunction of the index form is smooth up to the boundary. Precisely, the interior regularity is asserted in Corollary 7.11, Gilbarg-Trudinger [23], and the boundary regularity is stated in Theorem 5.4 and Section 8.2, LionsMagenes [41]. Then integrating by parts gives ∆f + (Ric(ν) + |A|2 )f + λf g dµ + − Σ ∂Σ ∂f + ∇ν ν, η f g ds = 0. ∂η Equivalently f solves the following Robin-type boundary value problem: ∆f + Ric(ν) + |A|2 f = −λf ∂f ∂η + ∇ν ν, η f = 0 in Σ on ∂Σ. Standard arguments for self-adjoint operators (Section 8.12, [23]) asserts that the eigenvalues are λ1 < λ2 ≤ · · · ≤ λi ≤ · · · and the eigenvectors {fi } form an orthogonal basis for W 1,2 (Σ). 62 3.6. Topology of minimal surfaces of low index Let h be an eigenfunction of λ1 . By the variational characterization of eigenfunctions, we have λ1 = min W 1,2 (Σ) I(f, f ) I(h, h) = . (f, f ) (h, h) But |h| will satisfy the same equality and hence is also an eigenfunction of λ1 . It follows that |h| is a nonnegative solution to the above boundary value problem. Then the Harnack inequality (Corollary 8.21, [23]) implies that |h| is indeed positive in the interior of Σ. Moreover, by Hopf’s Lemma and using the boundary condition, we get that |h| is positive on ∂Σ (Lemma 3.4 , [23]). Therefore h has a fixed sign on Σ. 3.6.2 A stability inequality for index-1 minimal surfaces Suppose the index of a minimal surface is 1. Then any eigenfunction which is not proportional to h has nonnegative eigenvalue. Since their spanning space in W 1,2 (Σ) is the orthogonal complement of h, this implies I(f, f ) ≥ 0, for all f orthogonal to h: (f, h) = Σ f h dµ = 0. From the above discussion, let h > 0 be a first eigenfunction. We want to use a specific function f orthogonal to h and containing information about the topology of Σ in the second variation formula (index form) above. Using arguments as in [26], [40] page 274 we have the following: Lemma 3.6.1. There exists a conformal map f : Σ → S 2 such that = 0 and f has degree ≤ [ g+3 2 ]. Σ f h dµ Proof. By gluing a disk on each boundary component of Σ, we may view Σ ¯ of genus g. There exists a conformal as a domain in a compact surface Σ map from this closed surface to the sphere ¯ → S2 ψ:Σ of degree ≤ [ g+3 2 ] (see [19]). Let G be the group of conformal diffeomorphisms of S 2 . We claim there exists ϕ ∈ G such that (ϕ ◦ ψ) h dµ = 0. Σ To see this, recall that the conformal transformation group G contains a subgroup which is homeomorphic to B 3 . That is, given a ∈ B 3 that is not the origin, let θ(a) = a/|a| ∈ S 2 , and let ϕ(t) be the one parameter family of conformal transformations of the ball B 3 that are dilations on the sphere 63 3.6. Topology of minimal surfaces of low index fixing the opposite poles θ(a) and −θ(a). In the group ϕ(t) there is a unique conformal automorphism ϕa that maps the origin to a. Define H : B 3 → B 3 by 1 (ϕa ◦ ψ)h dµ. H(a) = h Σ dµ Σ As a approaches the boundary ∂B 3 , ϕa (S 2 \ {−a}) → a and so (ϕa ◦ ψ) dµ → a h dµ. Σ Σ Therefore, H extends continuously to a map H : B 3 → B 3 which is the identity on ∂B 3 . By a standard argument in topology, H must be surjective. Therefore there exists a ∈ B 3 such that H(a) = 0, as claimed. Now let fi be the component functions of f that are orthogonal to h. We have |∇fi |2 − Ric(ν) + |A|2 fi2 dµ + I(fi , fi ) = Σ ∇ν ν, η fi2 ds ≥ 0. ∂Σ Summing over i, and using 3 2 i=1 |fi | |∇f |2 − Ric(ν) + |A|2 Σ = 1, we get ∇ν ν, η ds ≥ 0. dµ + ∂Σ ¯ → S 2 is conformal, Since f : Σ |∇f |2 dµ < Σ |∇f |2 dµ = 2Area(f (Σ)) = 2Area(S 2 )d(f ) ≤ 8π Σ g+3 . 2 Therefore (Ric(ν) + |A|2 ) dµ < 8π Σ 3.6.3 g+3 + 2 ∇ν ν, η ds. ∂Σ Controlling topology, area estimates, and rigidity for minimal surfaces of low index Now we prove Theorem 1.0.3. 64 3.6. Topology of minimal surfaces of low index Proof of theorem 1.0.3. Let Σ be a solution to the free boundary problem (Σ, ∂Σ) → (N, ∂N ). Choose a local orthonormal frame {e1 , e2 , e3 } along Σ such that e1 = T is the positively oriented unit tangent vector and e2 = η is the outward unit conormal along ∂Σ, and e3 = ν is the globally defined unit normal to Σ. For 1 ≤ i < j ≤ 3, let Rijij denote the sectional curvature of N for the section ei ∧ ej . Let R = R1212 + R1313 + R2323 be the scalar curvature of N , and let R33 = R1313 + R2323 (3.6.1) be the Ricci curvature for e3 = ν. Let K denote the Gauss curvature of Σ. From the Gauss equation and the fact that Σ is minimal we have 1 K = R1212 − |A|2 . 2 (3.6.2) First we assume that Σ has index 1. From the above, we have (R33 + |A|2 ) dµ < 8π Σ g+3 + 2 ∇ν ν, η ds. (3.6.3) ∂Σ Now we prove the three parts of the theorem. Part (i) Ric(N ) ≥ 0 and ∂N is weakly convex. From (3.6.1) and (3.6.2) we have R33 + 2K = R11 + R22 − |A|2 . (3.6.4) Inserting (3.6.4) into (3.6.3), we get (R11 + R22 − 2K) dµ < 8π Σ g+3 + 2 ∇ν ν, η ds. (3.6.5) ∂Σ By the Gauss-Bonnet theorem, kg ds = 2πχ(Σ) = 2π(2 − 2g − k), K dµ + Σ ∂Σ where kg is the geodesic curvature of ∂Σ in Σ. Recall that kg = − ∇T T, η , so K dµ − ∇T T, η ds = 2π(2 − 2g − k). (3.6.6) Σ ∂Σ 65 3.6. Topology of minimal surfaces of low index Inserting (3.6.6) into (3.6.5), and using the assumption that Ric(N ) ≥ 0, we get g+3 + 2 4π(2g + k − 2) < 8π ∇ν ν, η ds + 2 ∂Σ ∇T T, η ds. ∂Σ Since η is orthogonal to ∂N and ∂N is weakly convex we get g+ Since g+3 k −1< . 2 2 g g + 3 − 1+(−1) g+3 2 = , 2 2 it follows 1 + (−1)g < 5. 2 From this we obtain i) g + k ≤ 3 if g is even, ii) g + k ≤ 4 if g is odd. g+k+ Part (ii) R ≥ 0 and ∂N is weakly mean convex. Adding (3.6.1) and (3.6.2), we have 1 R33 + K = R − |A|2 . 2 (3.6.7) Inserting (3.6.7) into (3.6.3), we obtain 1 g+3 (R − K + |A|2 ) dµ < 8π + 2 2 Σ ∇ν ν, η ds. (3.6.8) ∂Σ Then using the nonnegative scalar curvature assumption and (3.6.6), we get −2π(2 − 2g − k) < 8π g+3 + 2 ∇ν ν, η ds + ∂Σ ∇T T, η ds. ∂Σ Since ∂N is weakly mean convex we obtain g+ g+3 k −1<2 . 2 2 Then 1 + (−1)g k −1<g+3− . 2 2 From this we obtain i) k ≤ 5 if g is even, ii) k ≤ 7 if g is odd. g+ 66 3.6. Topology of minimal surfaces of low index We now assume that Σ is stable; that is, I(f, f ) ≥ 0 for all f ∈ W 1,2 (Σ). Taking f to be a constant function, we obtain (R33 + |A|2 ) dµ ≤ Σ ∇ν ν, η ds. ∂Σ Using (3.6.7) we get 1 (R − K + |A|2 ) dµ ≤ 2 Σ ∇ν ν, η ds. ∂Σ By (3.6.6) we get Σ 1 R + |A|2 dµ − 2π(2 − 2g − k) ≤ 2 ∇ν ν, η ds + ∂Σ ∇T T, η ds. ∂Σ (3.6.9) Then since R ≥ 0 and ∂N is weakly mean convex, we get g+ k − 1 ≤ 0. 2 Therefore the only possibilities for (g, k) are (0, 1) or (0, 2) and Σ must be a disk or a cylinder. If Σ is a cylinder, from the above we must have R = 0 and |A|2 = 0 on Σ, and I(1, 1) = 0. Therefore f = 1 satisfies the Jacobi equation ∆f + (Ric(ν) + |A|2 )f = 0. This implies Ric(ν) = 0, and then from (3.6.7), K = 0. Hence Σ is a totally geodesic flat cylinder. We remark that this result can be viewed as the free boundary analogue to results of Schoen and Yau for compact minimal surfaces in compact ambient manifolds (Theorem 5.1, [50]), and Fischer-Colbrie and Schoen for complete minimal surfaces in complete ambient manifolds (Theorem 3, [21]). Part (iii) We now derive the area estimates. If Σ has index 1, from (3.6.8) and (3.6.6) we get R0 · Area(Σ) ≤ 8π and so Area(Σ) ≤ g+3 1 k − g+ −1 2 2 2 , 2π(7 − (−1)g − k) . R0 67 3.6. Topology of minimal surfaces of low index If Σ is stable, from (3.6.9) we get R0 · Area(Σ) ≤ 4π − g + k −1 2 . Then (g, k) has to be (0, 1) and Σ is a disk. Therefore, Area(Σ) ≤ 2π . R0 This completes the proof of Theorem 1.0.3. Remark. Let N be a compact orientable 3-manifold with boundary ∂N = ∅. Theorem 1.0.2 and 1.0.3 imply that if there exists a continuous map from a compact bordered Riemann surface with g = 0 and k ≥ 3 or g ≥ 1 and k ≥ 1, satisfying the incompressible assumption in Theorem 1.0.2, then N admits no metric of nonnegative scalar curvature, for which ∂N is weakly mean convex. In particular, this assumption is satisfied when π1 (N ) contains a subgroup abstractly isomorphic to the fundamental group of a compact Riemann surface with g ≥ 1 and k ≥ 1, or g = 0 and k ≥ 3, and π1 (N, ∂N ) = 0. Precisely, the first condition asserts the existence of a compact Riemann surface of the same topological type in N , which is incompressible. The second condition can be used to deform each boundary circle of this surface to some loop in ∂N by adding a cylinder to the surface. Thus we obtain a compact Riemann surface in N with boundary in ∂N satisfying the incompressible condition. 68 Bibliography [1] W. Abikoff, Degenerating families of Riemann surfaces, Ann. of Math. (2) 105 (1977), 29–44. [2] W. Abikoff, The Real Analytic Theory of Teichm¨ uller Space, Lecture Notes in Mathematics 820, Springer, Berlin, 1980. [3] S. Agmon, The Lp approach to the Dirichlet problem. I. Regularity theorems, Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 405–448. [4] G. Barles, S. Biton, and O. Ley, Uniqueness for parabolic equations without growth condition and applications to the mean curvature flow in R2 , J. Differential Equations 187 (2003), 456–472. [5] L. Bers, Topics in the real analytic theory of Teichm¨ uller space, Mimeographed notes, Univ. of Illinois at Urbana-Champaign. [6] A. Chau, J. Chen, and W. He, Lagrangian mean curvature flow for entire Lipschitz graphs, Calc. Var. Partial Differential Equations 44 (2012), no. 1-2, 199-220. [7] A. Chau, J. Chen, and W. He, Entire self-similar solutions to Lagrangian mean curvature flow, arXiv:0905.3869. [8] A. Chau, J. Chen, and Y. Yuan, Lagrangian mean curvature flow for entire Lipschitz graphs II, to appear in Math. Ann. DOI 10.1007/s00208013-0897-2. [9] A. Chau, J. Chen, and Y. Yuan, Rigidity of entire self-shrinking solutions to curvature flows, J. Reine Angew. Math. 664 (2012), 229–239. [10] J. Chen, A. Fraser, and C. Pang, Minimal immersions of compact bordered Riemann surfaces with free boundary, to appear in Trans. Amer. Math. Soc., arXiv:1209.1165. [11] J. Chen and C. Pang, Uniqueness of unbounded solutions of Lagrangian mean curvature flow, C. R. Math. Acad. Sci. Paris 347 (2009), 1031– 1034. 69 Bibliography [12] T. H. Colding and W. P. Minicozzi II, A course in minimal surfaces, Graduate Studies in Mathematics, 121, American Mathematical Society, Providence, RI, 2011. [13] M. G. Crandall, H. Ishii, and P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27 (1992), 1–67. [14] U. Dierkes, S. Hildebrandt, and A. J. Tromba, Global analysis of minimal surfaces, Revised and enlarged second edition. Grundlehren der Mathematischen Wissenschaften, 341. Springer, Heidelberg, 2010. [15] C. J. Earle and J. Eells, A fibre bundle description of Teichmller theory, J. Differential Geometry 3 (1969) 19–43. [16] C. J. Earle and A. Schatz, Teichm¨ uller theory for surfaces with boundary, J. Differential Geometry 4 (1970), 169–185. [17] K. Ecker and G. Huisken, Mean curvature evolution of entire graphs, Ann. of Math. 2 130 (1989), no.3, 453–471. [18] A. El Soufi and S. Ilias, Majoration de la seconde valeur propre d’un op´erateur de Schr¨ odinger sur une vari´et´e compacte et applications, J. Funct. Anal. 103 (1992), no. 2, 294–316. [19] H. M. Farkas and I. Kra, Riemann surfaces, Graduate Texts in Mathematics, 71, Springer, New York, 1992. [20] A. Fischer and A. Tromba, On a purely ”Riemannian” proof of the structure and dimension of the unramified moduli space of a compact Riemann surface, Math. Ann. 267 (1984), no. 3, 311–345. [21] D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), no. 2, 199–211. [22] A. Fraser, On the free boundary variational problem for minimal disks, Comm. Pure Appl. Math. 53 (2000), 931–971. [23] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics. Springer-Verlag, Berlin, 2001. [24] R. D. Gulliver, II, R. Osserman, and H. L. Royden, A theory of branched immersions of surfaces, Amer. J. Math. 95 (1973), 750–812. 70 Bibliography [25] R. Harvey and B. Lawson Jr, Calibrated geometries Acta Math. 148 (1982), 47-157. [26] J. Hersch, Quatre propri´et´es isop´erim´etriques de membranes sph´eriques homog`enes, C. R. Acad. Sci. Paris Sr. A-B 270 (1970), A1645–A1648. [27] R. A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985. [28] D. Husemoller, Fibre bundles, Third edition. Graduate Texts in Mathematics, 20. Springer-Verlag, New York, 1994. [29] J. Jost, Conformal mappings and the Plateau-Douglas problem in Riemannian manifolds, J. Reine Angew. Math. 359 (1985), 37-54. [30] J. Jost, Existence results for embedded minimal surfaces of controlled topological type, Ann. Sc. Norm. Sup. Pisa, Part I, 13 (1986), 15-5-0; Part II, 13 (1986), 401–426; Part III, 14 (1987), 165–167. [31] J. Jost, On the existence of embedded minimal surfaces of higher genus with free boundaries in Riemannian manifolds, Variational Methods for Free Surface Interfaces, ed P. Concus and R. Rinn, Springer, New York, 65–75. [32] J. Jost, Two-dimensional geometric variational problems, Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Ltd., Chichester, 1991. [33] L. Keen, Collars on Riemann surfaces, Discontinuous Groups and Riemann Surfaces, Ann. of Math. Studies 79 (1974), 263-268. [34] H. Koch and T. Lamm, Geometric flows with rough initial data, Asian J. Math. 16 (2012), 209-236. [35] E. Kuwert, A compactness result for loops with an H 1/2 -bound, J. Reine Angew. Math. 505 (1998), 1–22. [36] L. Lemaire, Boundary value problems for harmonic and minimal maps of surfaces into manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 9 (1982), no. 1, 91-103. [37] M. Li, A general existence theorem for embedded minimal surfaces with free boundary, arXiv:1204.2883. 71 Bibliography [38] M. Li, Width and rigidity of min-max minimal disks in three-manifolds with boundary, preprint. [39] M. Li, A rigidity theorem for the unit ball in R3 , preprint. [40] P. Li and S. T. Yau, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math. 69 (1982), no. 2, 269–291. [41] J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Vol. I. Translated from the French by P. Kenneth. Die Grundlehren der mathematischen Wissenschaften, Band 181. SpringerVerlag, New York-Heidelberg, 1972. [42] F. C. Marques and A. Neves, Rigidity of min-max minimal spheres in three-manifolds, arXiv:1105.4632. [43] W. Meeks III, L. Simon and S.-T. Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. (2) 116 (1982), no. 3, 621–659. [44] W. Meeks III and S.-T. Yau, Topology of three-dimensional manifolds and the embedding problems in minimal surface theory, Ann. of Math. (2) 112 (1980), no. 3, 441–484. [45] C. B. Morrey Jr., The problem of Plateau on a Riemannian manifold, Ann. of Math. (2) 49 (1948), 807–851. [46] D. Mumford, A remark on Mahler’s compactness theorem, Proc. Amer. Math. Soc. 28 (1971), 289–294. [47] A. Ros, One-sided complete stable minimal surfaces, J. Differential Geom. 74 (2006), no. 1, 69–92. [48] J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math. (2) 113 (1981), 1–24 [49] J. Sacks and K. Uhlenbeck, Minimal immersions of closed Riemann surfaces, Trans. Amer. Math. Soc. 271 (1982), 639–652. [50] R. Schoen and S. T. Yau, Existence of incompressible minimal surfaces and the topology of three dimensional manifolds with non-negative scalar curvature, Ann. of Math. 110 (1979), 127–142. 72 Bibliography [51] R. Schoen and S. T. Yau, Lectures on Differential Geometry, International Press, Cambridge, MA, 1994. [52] K. Smoczyk, Longtime existence of the Lagrangian mean curvature flow, Calc. Var. Partial Differential Equations 20 (2004), no. 1, 25-46. [53] K. Smoczyk and M. T. Wang, Mean curvature flow of Lagrangian submanifolds with convex potentianls, J. Differential Geom. 62 (2002), no. 2, 243–257. [54] M. Struwe, On a free boundary problem for minimal surfaces, Invent. Math. 75 (1984), no. 3, 547–560. [55] F. Tomi and A. J. Tromba, The index theorem for minimal surfaces of higher genus, Mem. Amer. Math. Soc. 117 (1995), no. 560. [56] S. T. Yau, Nonlinear analysis in geometry, Enseign. Math. 33 (1987), 109–158. [57] R. Ye, On the existence of area-minimizing surfaces with free boundary, Math. Z. 206 (1991), no. 3, 321–331. 73
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Uniqueness of Lagrangian mean curvature flow and minimal...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Uniqueness of Lagrangian mean curvature flow and minimal immersions with free boundary Pang, Chao 2013
pdf
Page Metadata
Item Metadata
Title | Uniqueness of Lagrangian mean curvature flow and minimal immersions with free boundary |
Creator |
Pang, Chao |
Publisher | University of British Columbia |
Date Issued | 2013 |
Description | In this thesis we investigate some problems on the uniqueness of mean curvature flow and the existence of minimal surfaces, by geometric and analytic methods. A summary of the main results is as follows. (i) The special Lagrangian submanifolds form a very important class of minimal submanifolds, which can be constructed via the method of mean curvature flow. In the graphical setting, the potential function for the Lagrangian mean curvature ow satisfies a fully nonlinear parabolic equation [formula omitted] where the ⋋j's are the eigenvalues of the Hessian D²u. We prove a uniqueness result for unbounded solutions of (1) without any growth condition, via the method of viscosity solutions ([4], [13]): for any continuous u₀ in ℝn, there is a unique continuous viscosity solution to (1) in ℝn x [0;∞). (ii) Let N be a complete, homogeneously regular Riemannian manifold of dimN ≥ 3 and let M be a compact submanifold of N. Let Ʃ be a compact Riemann surface with boundary. A branched immersion u : (Σ,∂Σ) → (N,M) is a minimal surface with free boundary in M if u(Σ) has zero mean curvature and u(Σ) is orthogonal to M along u(∂Σ)⊑ M. We study the free boundary problem for minimal immersions of compact bordered Riemann surfaces and prove that Σ if is not a disk, then there exists a free boundary minimal immersion of Σ minimizing area in any given conjugacy class of a map in C⁰(Σ,∂Σ;N,M) that is incompressible; the kernel of i* : π₁(M) → π₁(N) admits a generating set such that each member is freely homotopic to the boundary of an area minimizing disk that solves the free boundary problem. (iii) Under certain nonnegativity assumptions on the curvature of a 3-manifold N and convexity assumptions on the boundary M=∂N, we investigate controlling topology for free boundary minimal surfaces of low index: • We derive bounds on the genus, number of boundary components; • We prove a rigidity result; • We give area estimates in term of the scalar curvature of N. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2013-05-28 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NoDerivs 3.0 Unported |
DOI | 10.14288/1.0073861 |
URI | http://hdl.handle.net/2429/44511 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2013-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nd/3.0/ |
Aggregated Source Repository | DSpace |
Download
- Media
- 24-ubc_2013_fall_pang_chao.pdf [ 658.52kB ]
- Metadata
- JSON: 24-1.0073861.json
- JSON-LD: 24-1.0073861-ld.json
- RDF/XML (Pretty): 24-1.0073861-rdf.xml
- RDF/JSON: 24-1.0073861-rdf.json
- Turtle: 24-1.0073861-turtle.txt
- N-Triples: 24-1.0073861-rdf-ntriples.txt
- Original Record: 24-1.0073861-source.json
- Full Text
- 24-1.0073861-fulltext.txt
- Citation
- 24-1.0073861.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.24.1-0073861/manifest