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

主要研究者：孙永Alice Chang/ Paul Yang，Princeton UniversityDMS-0245266摘要：建议的分析部分涉及两个问题，涉及共形几何中的非线性微分方程。第一个是制定一个一般性的概念弱解的一个家庭的完全非线性方程包含方程规定的对称函数的Weyl-Schouten张量，并提供标准，消除这种方程的奇异性。这将开辟一条道路，找到解决这些方程的一个更传统的变分方法。第二个问题是在三维和四维的共形几何中找到四阶方程的Sobolev不等式。这个建议的几何部分是关于我们最近的工作在规定四维流形上的Weyl-Schouten张量的第二对称函数到一类四维流形的同构分类中的应用，以及在高维流形上Kleinian群研究中的应用.本文提出了求解一类非线性微分方程的新方法，与保形几何相关联，其中主要数据是角度测量的知识。解决这些方程的能力给了我们新的数值不变量，最终使我们能够将这些几何形状在三维和四维中分类，这是几何和拓扑界广泛关注的一个主题。微分方程是高度非线性的，出现在一个家庭这样的方程，最近才屈服于几何方法。这一发展将为结构非线性偏微分方程的分析提供一套新的工具，而结构非线性偏微分方程也是一个广受关注的课题。

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.