Fully nonlinear partial differential equations and related problems in geometry

摘要：本文工作的主要目的是理解一些由几何和分析问题引起的非线性椭圆方程。研究方向为欧几里得空间和双曲空间中超曲面的平台型问题、黎曼流形上由Ricci张量函数决定的完全共形度量、正曲率或负曲率二维黎曼流形的等长嵌入问题以及复杂的Monge-Ampre方程。这些问题涉及到解决一些完全非线性的偏微分方程。由于它们起源于几何，这些方程密切相关，因此有许多共同的技术问题。这个项目的进展可能会导致流形上的非线性偏微分方程理论和几何分析中其他问题的解决。几何分析和全非线性方程的理论和方法在理解纯数学和应用数学中的难题，如庞加莱猜想、广义相对论中的正质量定理和彭罗斯不等式，以及在图像处理、光学反射器设计、最优质量输运、数学生物学和物理学中的应用方面发挥着重要作用。该建议涉及微分几何和高度非线性偏微分方程中一些非常典型的问题。这项研究将促进对这些问题的理解，并开发有助于研究由数学和物理科学中的其他问题引起的完全非线性方程的技术。

Abstract:The primary goal of the work described in this proposal is to understand some nonlinear elliptic equations which arise from problems in geometry and analysis.The research will be focused on geometric problems in four areas:Plateau-type problems for hypersurfaces in Euclidean and hyperbolic spaces, complete conformal metrics determined by functions of Ricci tensor on Riemannian manifolds with boundary, problems in isometric embedding of 2-dimensional Riemannian manifolds of positive or negative curvature, and complex Monge-Ampre equations.These problems involve solving some fully nonlinear partial differential equations. Because of their origin in geometry, these equations are closely related and therefore share many common technical issues. Progress in this project may lead to solutions to others problems in the theory of nonlinear partial differential equations on manifolds and geometric analysis.The theory and methods of geometric analysis and fully nonlinear equations play important roles in understanding difficult problems in pure and applied mathematics, such as the Poincare conjecture, the positive mass theorem and Penrose inequality in general relativity, and applications in image processing, optical reflector designs, optimal mass transport, mathematical biology and physics.The proposal concerns some very typical problems in differential geometry and highly nonlinear partial differential equations.The research will advance the understanding of these problems and develop techniques that will be useful in studying fully nonlinear equations which arise from other problems in mathematics and physical sciences.