Non-linear Partial Differential Equations and Applications to Problems in Geometry

摘要：本文的解析部分讨论了由共形几何引起的非线性微分方程的两个问题。第一个是对一类完全非线性方程的弱解的一般概念，这些方程包含规定Weyl-Schouten张量的对称函数的方程，并提供消除此类方程奇点的准则。这将为用更传统的变分方法求解这些方程开辟道路。第二步是找到三维和四维共形几何中出现的四阶方程的Sobolev不等式。本建议的几何部分涉及我们最近在四维流形上的weyl - schouten张量的第二对称函数的规定在一类四维流形的微分同态分类中的应用，以及在高维流形中Kleinian群的研究中的应用。本文提出了一种新的方法来解决一类非线性微分方程，这些方程与保形几何有关，其中主要数据是角度测量的知识。求解这些方程的能力给我们提供了新的数值不变量，最终使我们能够在三维和四维中对这些几何图形进行分类，这是几何和拓扑学界广泛感兴趣的主题。微分方程是高度非线性的，并且出现在最近才屈服于几何方法的一类方程中。这种发展将产生一套新的工具来分析结构非线性偏微分方程，这也是一个广泛感兴趣的主题。

PI: Sun-Yung Alice Chang/ Paul Yang, Princeton UniversityDMS-0245266Abstract:The analytic part of the proposal is concerned with two questions concerning nonlinear differential equations arising from conformal geometry. The first is to formulate a general notion of weak solutions for a family of fully nonlinear equations containing the equations to prescribe the symmetric functions of the Weyl-Schouten tensor, and to provide criteria for removal of singularities of such equation. This would open the way to find solutions to these equations by a more traditional variational method. The second is to find Sobolev inequalities for fourth order equations that arise in conformal geometry in dimensions three and four. The geometric part of this proposal is concerned with applications of our recent work in prescribing the second symmetric functions of theWeyl-Schouten tensor on a four-dimensional manifold to the diffeomorphisms classification of a class of four-dimensional manifolds, as well as applications to the study of Kleinian groups in higher dimensional manifolds.This proposal is concerned with new methods to solve a family of nonlinear differential equations that is associated with conformal geometry in which the primary data is the knowledge of angle measurements. The ability to solve these equations gives us new numerical invariants that will eventually allow us to classify these geometries in three and four dimensions a subject that is of wide interest in the geometry and topology community. The differential equations are highly nonlinear and appear among a family of such equations that have only recently yielded to a geometric approach. Such development will generate a set of new tools to analyze the structure nonlinear partial differential equations a subject also of wide interest in general.