SurveysinDifferentialGeometryXVIII Preface Each year the Journal of Differential Geometry (JDG) sponsors a con- ference on Geometry and Topology. The conference is held every third year at Harvard University, and other years at Lehigh University. The current volume includes papers presented by several speakers at both the 2011 conference at Harvard and the 2012 conference at Lehigh. We havearticlesbySimonBrendle,ontheLagrangianminimalsurfaceequation and related problems; by Sergio Cecotti and Cumrun Vafa, concerning classification of complete N = 2 supersymmetric theories in 4 dimensions; by F. Reese Harvey and H. Blaine Lawson Jr., on existence, uniqueness, and removable singularities for non-linear PDEs in geometry; by Ja´nos Kolla´r, concerning links of complex analytic singularities; by Claude LeBrun, on Calabi energies of extremal toric surfaces; by Mu-Tao Wang, concerning mean curvature flows and isotopy problems; and by Steve Zelditch, on eigenfunctions and nodal sets. We are grateful to the many distinguished geometers and topologists who presented invited talks at these two conferences, especially those who contributed articles to this volume of the Surveys in Differential Geometry book series. Huai-Dong Cao Lehigh University Shing-Tung Yau Harvard University v SurveysinDifferentialGeometryXVIII Contents Preface v On the Lagrangian minimal surface equation and related problems Simon Brendle 1 Classification of complete N = 2 supersymmetric theories in 4 dimensions Sergio Cecotti and Cumrun Vafa 19 Existence, uniqueness and removable singularities for nonlinear partial differential equations in geometry F. Reese Harvey and H. Blaine Lawson, Jr. 103 Links of complex analytic singularities Ja´nos Kolla´r 157 Calabi energies of extremal toric surfaces Claude LeBrun 195 Mean curvature flows and isotopy problems Mu-Tao Wang 227 Eigenfunctions and nodal sets Steve Zelditch 237 vii SurveysinDifferentialGeometryXVIII On the Lagrangian minimal surface equation and related problems Simon Brendle Abstract. We give a survey of various existence results for minimal Lagrangian graphs. We also discuss the mean curvature flow for La- grangian graphs. 1. Background on minimal Lagrangian geometry Minimal submanifolds are among the central objects in differential geometry.Thereisanimportantsubclassofminimalsubmanifoldswhichwas introducedbyHarveyandLawson[6]in1982.GivenaRiemannianmanifold (M,g), a calibrating form Ω is a closed m-form on M with the property that Ω(e ,...,e ) ≤ 1 for each point p ∈ M and every orthonormal k-frame 1 m {e ,...,e } ⊂ T M.Anorientedm-dimensionalsubmanifoldΣ⊂ M issaid 1 m p to be calibrated by Ω if Ω(e ,...,e ) = 1 for every point p ∈ Σ and every 1 m positively ortiented orthonormal basis {e ,...,e } of T Σ. Using Stokes 1 m p theorem, Harvey and Lawson showed that every calibrated submanifold is necessarily minimal: Theorem 1.1 (R. Harvey, H.B. Lawson [6]). Let (M,g) be a Riemann- ian manifold. Moreover, let Ω be a calibrating k-form and let Σ be a k- dimensional submanifold calibrated by Σ. Then Σ minimizes volume in its homology class. In the following, we consider the special case when (M,g) is the Eu- clidean space R2n. We denote by (x1,...,xn,y1,(cid:2)...,yn) the standard co- ordinates on R2n. Moreover, we denote by ω = n dx ∧ dy the stan- k=1 k k dard symplectic form. Let J be the associated complex structure, so that J ∂ = ∂ and J ∂ = − ∂ . Finally, we define ∂x ∂y ∂y ∂x k k k k σ = (dx +idy )∧...∧(dx +idy ). 1 1 n n The author was supported in part by the National Science Foundation under grant DMS-0905628. (cid:2)c 2013InternationalPress 1 2 SIMONBRENDLE Note that σ is a complex-valued n-form on R2n. Moreover, we have σ(Jv ,v ,...,v ) = iσ(v ,v ,...,v ) 1 2 n 1 2 n for all vectors v ,...,v ∈ R2n. 1 n Let now Σ be a submanifold of R2n of dimension n. Recall that Σ is said to be Lagrangian if ω| = 0. If Σ is a Lagrangian submanifold, then it Σ can be shown that |σ(e ,...,e )| = 1, where {e ,...,e } is an orthonormal 1 n 1 n basis of T Σ. We may therefore write p (1) σ(e ,...,e ) = eiγ 1 n for some function γ : Σ → R/2πZ. The function γ is referred to as the Lagrangian angle of Σ. The mean curvature vector of a Lagrangian submanifold Σ is given by J∇Σγ, where ∇Σγ ∈ T Σ denotes the gradient of the Lagrangian angle. In p particular, this implies: Theorem 1.2 (R. Harvey, H.B. Lawson [6]). If Σ is a Lagrangian submanifold with H = 0, then the Lagrangian angle must be constant. Conversely, if Σ is a Lagrangian and the Lagrangian angle is constant (so that γ = c), then Σ is calibrated by the n-form Ω = Re(e−icσ). In particular, minimal Lagrangian submanifolds are special cases of calibrated submanifolds. The first non-trivial examples of minimal Lagrangian submanifolds in R2n were constructed by Harvey and Lawson [6]. These examples are nearly flat and are constructed by means of the implicit function theorem. 2. Minimal Lagrangian graphs in R2n We now assume that Σ is an n-dimensional submanifold of R2n which can be written as a graph over a Lagrangian plane in R2n. In other words, we write Σ = {(x ,...,x ,y ,...,y ) ∈ R2n : (y ,...,y ) = f(x ,...,x )}. 1 n 1 n 1 n 1 n Here, the map f is defined on some domain in Rn and takes values in Rn. The condition that Σ is Lagrangian is equivalent to the condition that ∂ f = ∂ f . Thus, Σ is Lagrangian if and only if the map f can locally k l l k be written as the gradient of some real-valued function u. In this case, the Lagrangian angle of Σ is given by (cid:3)n γ = arctan(λ ), k k=1 where λ ,...,λ denote the eigenvalues of Df(x) = D2u(x). Therefore, Σ 1 k is a minimal Lagrangian submanifold if and only if u satisfies the Hessian equation (cid:3)n (2) arctan(λ ) = c. k k=1 LAGRANGIAN MINIMAL SURFACE EQUATION 3 A natural question is to classify all entire solutions of (2). In this direction Tsui and Wang proved the following result: Theorem 2.1 (M.P. Tsui, M.T. Wang [15]). Let f : Rn → Rn be a smooth map with the property that Σ = {(x,f(x)) : x ∈ Rn} is a minimal Lagrangian graph. Moreover, we assume that, for each point x ∈ Rn, the eigenvalues of Df(x) satisfy λ λ ≥ −1 and |λ | ≤ K. Then f is an affine i j i function. A closely related Bernstein-type result was established independently in [23]: Theorem 2.2 (Y. Yuan [23]). Let u : Rn → R be a smooth convex solution of (2). Then u is a quadratic polynomial. In order to study the equation (2) on a bounded domain in Rn, one needs to impose a boundary condition. One possibility is to impose a Dirichlet boundary condition for the potential function u. This boundary value problem was studied in the fundamental work of Caffarelli, Nirenberg, andSpruck[4].Inparticular,theyobtainedthefollowingexistencetheorem: Theorem 2.3 (L. Caffarelli, L. Nirenberg, J. Spruck [4]). Let Ω be a uniformly convex domain in Rn, and let ϕ : ∂Ω → R be a smooth function. Then there exists a smooth function u : Ω → R satisfying (cid:3)n (cid:4)n−1(cid:5) arctan(λ ) = π k 2 k=1 and u| = ϕ. ∂Ω We now describe another natural boundary condition for (2). Instead of prescribing the boundary values of u, we prescribe the image of Ω under the map f = ∇u. This choice of boundary condition has been studied before in connection with the Monge-Amp`ere equation (see [3], [17], [18]). Theorem 2.4 (S. Brendle, M. Warren [2]). Let Ω and Ω˜ be uniformly convex domains in Rn. Then we can find a smooth function u : Ω → R and a real number c with the following properties: (i) The function u is uniformly convex. (ii) The function u solves the equation (2). (iii) The map ∇u : Ω → R is a diffeomorphism from Ω to Ω˜. Moreover, the pair (u,c) is unique. Thus, we can draw the following conclusion: Corollary 2.5 (S. Brendle, M. Warren [2]). Let Ω and Ω˜ be uniformly convex domains in Rn with smooth boundary. Then there exists a diffeomor- phism f : Ω → Ω˜ such that the graph Σ = {(x,f(x)) : x ∈ Ω} is a minimal Lagrangian submanifold of R2n. 4 SIMONBRENDLE In particular, the submanifold Σ satisfies ∂Σ ⊂ ∂Ω × ∂Ω˜. Thus, the surface Σ satisfies a free boundary value problem. Wenotethatthepotentialfunctionuisnotageometricquantity;onthe other hand, the gradient ∇u = f does have geometric significance. From a geometric point of view, the second boundary value problem is more natural than the Dirichlet boundary condition. We now describe the proof of Theorem 2.4. The uniqueness statement followsfromastandardargumentbasedonthemaximumprinciple.Inorder to prove the existence statement, we use the continuity method. The idea is to deform Ω and Ω˜ to the unit ball in Rn. As usual, the central issue is to bound the Hessian of the potential function u. In geometric terms, this corresponds to a bound on the slope of Σ. Proposition 2.6 ([2]). Let us fix two uniformly convex domains Ω and Ω˜. Moreover, let u be a convex solution of (2) with the property that ∇u is a diffeomorphism from Ω to Ω˜. Then |D2u(x)| ≤ C for all points x ∈ Ω and all vectors v ∈ Rn. Here, C is a positive constant, which depends only on Ω and Ω˜. The proof of Proposition 2.6 is inspired by earlier work of Urbas on the Monge-Amp`ere equation. By assumption, we can find uniformly convex boundary defining functions h : Ω → (−∞,0] and h˜ : Ω˜ → (−∞,0], so that h| = 0 and h˜| = 0. Moreover, let us fix a constant θ > 0 such that ∂Ω ∂Ω˜ D2h(x) ≥ θI for all points x ∈ Ω and D2h˜(y) ≥ θI for all points y ∈ Ω˜. In the following, we sketch the main steps involved in the proof of Proposition 2.6. Step 1: Let u be a convex solution of (2) with the property that ∇u is a diffeomorphism from Ω to Ω˜. Differentiating the equation (2), we obtain (cid:3)n (3) a (x)∂ ∂ ∂ u(x) = 0 ij i j k i,j=1 for all x ∈ Ω and all k ∈ {1,...,n}. Here, the coefficients a (x) are defined ij as the components of the matrix A(x) = (I +(D2u(x))2)−1. We now define a function H : Ω → R by H(x) = h˜(∇u(x)). Using the identity (3), one can show that (cid:6) (cid:6) (cid:6) (cid:3)n (cid:6) (cid:6)(cid:6) aij(x)∂i∂jH(x)(cid:6)(cid:6) ≤ C i,j=1 for some uniform constant C. Using the maximum principle, we conclude that H(x) ≥ Ch(x) for all points x ∈ Ω. Here, C is a uniform constant whichdependsonlyonΩandΩ˜.Thisimplies(cid:10)∇h(x),∇H(x)(cid:11) ≤ C|∇h(x)|2 at each point x ∈ ∂Ω. As a result, we can bound certain components of the Hessian of u along ∂Ω. LAGRANGIAN MINIMAL SURFACE EQUATION 5 Step 2: In the next step, we prove a uniform obliqueness estimate. To that end, we consider the function χ(x) = (cid:10)∇h(x),∇h˜(∇u(x))(cid:11). It is not difficult to show that χ(x) > 0 for all x ∈ ∂Ω. The goal is to obtain a uniform lower bound for infx∈∂Ωχ(x). Using the relation (3), one can show that (cid:6) (cid:6) (cid:6) (cid:3)n (cid:6) (cid:6)(cid:6) aij(x)∂i∂jχ(x)(cid:6)(cid:6) ≤ C i,j=1 for some uniform constant C. We can therefore find a uniform constant K such that (cid:3)n a (x)∂ ∂ (χ(x)−Kh(x)) ≤ 0. ij i j i,j=1 We now consider a point x ∈ ∂Ω, where the function χ(x)−Kh(x) attains 0 its global minimum. Then ∇χ(x ) = (K −μ)∇h(x ) for some real number 0 0 μ ≥ 0. Hence, we obtain (K −μ)χ(x ) = (cid:10)∇χ(x ),∇h˜(∇u(x ))(cid:11) 0 0 0 (cid:3)n = ∂ ∂ h(x )(∂ h˜)(∇u(x ))(∂ h˜)(∇u(x )) i j 0 i 0 j 0 i,j=1 (cid:3)n + (∂ ∂ h˜)(∇u(x ))∂ h(x )∂ H(x ) i j 0 i 0 j 0 i,j=1 (cid:3)n ≥ θ|∇h˜(∇u(x ))|2+ (∂ ∂ h˜)(∇u(x ))∂ h(x )∂ H(x ). 0 i j 0 i 0 j 0 i,j=1 Since ∇H(x ) is a positive multiple of ∇h(x ), it follows that 0 0 Kχ(x ) ≥ θ|∇h˜(∇u(x ))|2. 0 0 Since infx∈∂Ωχ(x) = χ(x0), we obtain a uniform lower bound for infx∈∂Ωχ(x). Step 3: Having established the uniform obliqueness estimate, we next bound the tangential components of the Hessian D2u(x) for each point x ∈ ∂Ω. To explain this, let (cid:7) (cid:8) (cid:3)n M = sup ∂ ∂ u(x)z z : x ∈ ∂Ω, z ∈ T (∂Ω), |z| = 1 . k l k l x k,l=1 Our goal is to establish an upper bound for M. To that end, we fix a point x ∈ ∂M and a vector w ∈ T (∂Ω) such that |w| = 1 and 0 x0 (cid:3)n ∂ ∂ u(x )w w = M. k l 0 k l k,l=1