Higher Order Elliptic Operators and Applications to Problems in Conformal Geometry

Chang建议继续她在共形几何中涉及高阶椭圆算子和系统的问题的研究。本文的主题是通过一个四阶椭圆算子研究高斯-博内公式在4流形上的可积性，该算子的第一项是双拉普拉斯算子，称为Paneitz算子。提出了三个研究项目。第一个将被积函数的研究与蒙日-安培方程的研究联系起来。第二个是将她早期关于双调和映射的规律性的工作扩展到一般情况下。第三是推广Cohn-Vossen和Huber的经典工作，将高斯曲率的增长与欧拉特征联系起来，从完全曲面到完全4流形。该项目的主要动机是通过解析方法研究流形的几何和拓扑结构——主要是通过研究流形曲率满足的偏微分方程。

Chang proposes to continue her study of problems in conformal geometry involving higher order elliptic operators and systems. The main theme is to study the integrand of the Gauss-Bonnet formula on 4-manifolds via a fourth order elliptic operator with leading term the bi-Laplace operator called the Paneitz operator. Three research projects are proposed. The first one connects the study of the integrand to the study of Monge-Ampere equations. The second one is to extend her earlier work on the regularity of bi-harmonic maps to a general setting. The third one is to generalize the classical work of Cohn-Vossen and Huber relating the growth of the Gaussian curvature to the Euler characteristic from complete surfaces to complete 4-manifolds. The main motivation of the project is to study the geometric and topological structure of manifolds via analytic methods--mainly via the study of the partial differential equations satisfied by the curvature of the manifold.