Fully Nonlinear Elliptic and Parabolic Equations in Differential Geometry

NSF提案DMS-0204590田纳西大学首席研究员关博标题：微分几何中的完全非线性椭圆型和抛物型方程摘要：完全非线性椭圆型和抛物型方程是由微分几何中的许多问题引起的。近年来，这些方程引起了人们的极大关注，人们对这些方程及其相关几何问题的理解也取得了重大进展。在这个项目中，首席调查员将继续朝这个方向进行研究。本课题要研究的问题包括：非负曲率度量的等距嵌入；欧氏空间和更一般的黎曼流形中带边界的非负常高斯曲率超曲面的问题，包括存在性、唯一性和正则性；Minkowski空间中常Gauss曲率的类空全图；求指定Weingarten曲率的闭凸超曲面的Minkowski问题；非凸域上退化的Monge-Ampere方程解的正则性；Kahler流形中多复Green函数的正则性和全纯函数的存在性；黎曼流形上的Hessian方程及其在几何问题中的应用；常曲率平均双曲空间(或Weingarten超曲面)中的超曲面以及超曲面的曲率函数演化。这些问题中的大多数方程都是高度非线性的。这些方程还模拟了科学和工程中的各种现象。求解这类方程在很大程度上取决于建立高达二阶导数的先验估计。对于本项目中提出的许多问题，仅靠这些估计往往不足以导致解的存在；还有来自几何和分析的其他障碍。这些都提出了一些具有挑战性的问题。对这些问题的研究还可以发展出在工程和科学上有用的解的数值逼近方法。

NSF proposal DMS - 0204590Principal Investigator: Bo Guan, University of Tennessee Title: FULLY NONLINEAR ELLIPTIC AND PARABOLIC EQUATIONS IN DIFFERENTIAL GEOMETRYAbstract:Fully nonlinear elliptic and parabolic equations arise from many problemsin differential geometry. In recent years these equations have attracted a lot of attention and significant progresses have been made to understandthese equations and related geometric problems. In this project, the principalinvestigator will continue his research in this direction. The problems to be investigated in this project include isometric embeddings of metrics of nonnegative curvature; questions about hypersurfaces of nonnegative constant Gauss curvature with boundary in Euclidean space and more general Riemannian manifolds, including existence, uniqueness and regularity; spacelike entire graphs of constant Gauss curvature in Minkowski space; Minkowski type problemsof finding closed convex hypersurfaces of prescribed Weingarten curvatures;regularity of solutions to degenerate Monge-Ampere equations in non-convex domains; regularity of pluricomplex Green functions and existence of holomorphic functions in Kahler manifolds; Hessian equations on Riemannian manifolds and applications in geometric problems; hypersurfaces in hyperbolic space of constant mean curvature (or Weingarten hypersurfaces) with prescribedasymptotic boundary at infinity; and evolution of hypersurfaces by curvature functions.Equations arising from most of these problems are highly nonlinear. These equations also model various phenomena in sciences and engineering. Solving such equations heavily depends on establishing apriori estimates up to second order derivatives. For many of the proposed problems in this project, these estimates alone are often not enough to lead to existence of solutions; there are other obstructions from geometry and analysis. These all impose challenging questions. Research on these problems may also develop methods of numerical approximations to the solutions that are useful in engineering andscience.