Fully nonlinear partial differential equations and related problems in geometry

文摘：本论文的主要目标是研究几何和分析中的非线性椭圆方程，主要研究四个方面的几何问题：欧氏空间和双曲空间中超曲面的Plateau型问题，黎曼流形上Ricci张量函数确定的完全共形度量，正曲率或负曲率的二维黎曼流形的等距嵌入问题，以及复Monge-Ampre方程，这些问题涉及到求解完全非线性偏微分方程。由于它们起源于几何学，这些方程密切相关，因此具有许多共同的技术问题。该项目的进展可能会导致流形上非线性偏微分方程理论和几何分析中其他问题的解决，几何分析和完全非线性方程的理论和方法在理解理论和应用数学中的难题，如Poincare猜想，广义相对论中的正质量定理和Penrose不等式，以及在图像处理中的应用方面发挥重要作用。光学反射镜设计、最优质量输运、数学生物学和物理学等领域的研究，并涉及微分几何和高度非线性偏微分方程中的一些非常典型的问题，这些研究将促进对这些问题的理解，并发展出研究由数学和物理学中的其他问题所产生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.