Fully Nonlinear Elliptic and Parabolic Equations in Differential Geometry

题目：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产生于微分几何中的许多问题。近年来，这些方程引起了人们的广泛关注，并在理解这些方程和相关几何问题方面取得了重大进展。在这个项目中，首席研究员将继续这个方向的研究。本课题研究的问题包括：非负曲率度量的等距嵌入；欧几里德空间中非负常高斯曲率带边界的超曲面和更一般的黎曼流形的存在性、唯一性和正则性问题；闵可夫斯基空间中具有常高斯曲率的类空间全图；求给定Weingarten曲率的闭凸超曲面的Minkowski型问题退化Monge-Ampere方程解在非凸区域的正则性复数复数Green函数的正则性与Kahler流形中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平均曲率双曲空间中在无穷远处具有规定渐近边界的超曲面（或Weingarten超曲面）以及曲率函数对超曲面的演化。由这些问题产生的方程大多是高度非线性的。这些方程还可以模拟科学和工程中的各种现象。求解这类方程在很大程度上依赖于建立二阶导数的先验估计。对于本项目中提出的许多问题，仅凭这些估计往往不足以得出解决方案；还有其他来自几何和分析的障碍。这些都提出了具有挑战性的问题。对这些问题的研究也可以发展出在工程和科学中有用的数值近似解的方法。

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.