We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge- Amp`ere type. These two problems are: the local isometric embedding problem for two-dimensional Riemannian manifolds, and the problem of locally prescribed Gaussian curvature for surfaces in R3. We prove a general local existence result for a large class of Monge-Amp`ere equations in the plane, and obtain as corollaries the existence of regular solutions to both problems, in the case that the Gaussian curvature vanishes to arbitrary finite order on a single smooth curve.

This article studies the mean curvature flow of Lagrangian submanifolds. In particular, we prove the following global existence and convergence theorem: if the potential function of a Lagrangian graph in $T^{2n}$ is convex, then the flow exists for all time and converges smoothly to a flat Lagrangian submanifold. We also discuss various conditions on the potential function that guarantee global existence and convergence.

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We study the existence of special Lagrangian submanifolds of log Calabi–Yau manifolds equipped with the complete Ricci-flat Kähler metric constructed by Tian and Yau. We prove that if X is a Tian–Yau manifold and if the compact Calabi–Yau manifold at infinity admits a single special Lagrangian, then X admits infinitely many disjoint special Lagrangians. In complex dimension 2, we prove that if Y is a del Pezzo surface or a rational elliptic surface and D∈|-KY| is a smooth divisor with D^2=d , then X=Y\D admits a special Lagrangian torus fibration, as conjectured by Strominger–Yau–Zaslow and Auroux. In fact, we show that X admits twin special Lagrangian fibrations, confirming a prediction of Leung and Yau. In the special case that Y is a rational elliptic surface or Y=P^2, we identify the singular fibers for generic data, thereby confirming two conjectures of Auroux. Finally, we prove that after a hyper-Kähler rotation, X can be compactified to the complement of a Kodaira type I_d fiber appearing as a singular fiber in a rational elliptic surface π: Y →P^1.