Monge-Ampère Equations In Geometry

Monge-Ampère equation is a very important fully nonlinear second-order partial differ-ential equation.It naturally arises from,among others,Weyl and Minkowski problems in classical geometry and Calabi conjecture in Kahler geometry.Indeed,the Complex Monge-Ampère equation plays central roles complex geometry and analysis,while real Monge-Ampère equations have close connections to the optimal transportation problem,geometric optics,conformal geometry and affine geometry,etc..The real Monge-Ampère equations on a Riemannian manifold(M,g)has the following form:det((?)2u + ?)=?—det g,(3)where ? is a(0,2)tensor.Similarly,the complex Monge-Ampère equation can be defined on a general Hermitian manifold(M,?)These are fully nonlinear partial differential equations.We are mainly concerned with the elliptic case,for which one applies the classical continuity method to show the existence of solutions.A crucial step for the continuity method is to establish C2,' estimate,by Evans-Krylov theorem,we are only need to establish C0,C1,2 estimates.The article is organized as follows:In Section 1,we introduce Monge-Ampère equations of classical geometry and derived the corresponding Monge-Ampère equation of several clas-sical problem(isometric embedding,prescribed Gaussian curvature,the optimal transporta-tion problem,Minkowski problem),that according with the general equation(3).In Section 2,we introduce complex Monge-Ampère equation.We focus on Yau's solution,the crucial part is to establish zeroth-order estimate,first and second-order estimate can be controlled by zero-order estimate.With the aid of Evans-Krylov and Schauder estimates,the existence of the solution is proved using the continuity method.