Non-linear Problems in Geometry

[703154]提出的研究涉及黎曼和微分几何领域的一些几何问题。曲率流的渐近形状、局部二维等距嵌入问题、Stein流形上的复格林函数及解析函数的存在性、Cartan-Hadamard流形上的等距不等式等问题是研究的重点。使用的主要工具是蒙日-安培方程理论。利用非线性椭圆型偏微分方程解决微分几何中的基本问题是现代数学的伟大成就之一。也许最重要的非线性椭圆方程是蒙日-安培方程，在过去的几年里，我们对这个方程的理解有了重大的进展。偏微分方程模拟各种自然现象，其中几个参数同时变化，并且可以观察到它们在短时间间隔内的变化率。到目前为止，这种方程的一般理论非常缺乏，特别是当方程是完全非线性的时候。

9703154 Spruck The proposed research deals with a number of geometric problems in the area of Riemannian and differential geometry. Asymptotic shape of curvature flows, local two dimensional isometric embedding problem, complex Green's function and existence of analytic functions on Stein manifolds, and isoperimetric inequality on Cartan-Hadamard manifolds are among the problems to be pursued. The main tool to be used is the theory of Monge-Ampere equations. The use of nonlinear elliptic partial differential equations to solve fundamental problems in differential geometry has been one of the great achievements of modern mathematics. Perhaps the most important nonlinear elliptic equation is the Monge-Ampere equation and there have been significant advances in our understanding of this equation in the last several years. Partial differential equations model various natural phenomena where several parameters vary simultaneously and their rates of change over a short time interval can be observed. Thus far, a general theory of such equations is very much lacking, especially when the equation is fully nonlinear.