Gemetric Structures on Manifolds and Global Analysis

在流形的众多几何结构中，我们主要研究与共形几何密切相关的结构。以下是本研究项目的主要成果：1.标量曲率方程。这个方程描述了黎曼度量的共形变化下的标量曲率。对非紧流形进行了系统的分析，明确了具有指定数量曲率的完备共形度量空间. Weyl结构。这是一个挠自由仿射联络，它与给定的共形类相容。证明了Ricci曲率是Weyl结构的完全不变量。并对紧致流形上的共形平坦Einstein-Weyl结构进行了分类. Moebius几何。在球面上给定拓扑类型的正则闭曲线至多有五条自相交线的情况下，完全确定了该曲线的最少顶点数。此外，我们还引入了正则曲线的Schwarzian导数。这导致了Nehari型内射性结果的新证明。其要点是流形的共形结构诱导出流形上正则曲线的可积射影结构。证明了曲线的射影展开映射的内射性蕴涵了浸入到Moebius空间的内射性。

Among many geometric structures of a manifold we are mainly interested in those structures which are closely related to the conformal geometry. Here are some of main results of this research project :1. The scalar curvature equation. This equation describes the scalar curvature under a conformal change of a Riemannian metric. A systematic analysis has been done on non-compact manifolds, and the space of complete confomal metrics with prescribed scalar curvature is made clearer.2. The Weyl structure. This is a torsion free affine connection that is compatible with a given conformal class. It is shown that the Ricci curvature is a complete invariant of a Weyl structure. Also conformally flat Einstein-Weyl structures on compact manifolds are classified.3. Moebius geometry. The minimum number of vertices of a regular closed curve on the sphere with given topological type is completely determined in the case when the curve has at most five self-inter-sections. Also we introduce a Schwarzian derivative of a regular curve. This leads to new proofs of injectivity results of Nehari type. A gist is that a confomal strucutre of a manifold induces an integrable projective structure of a regular curve on the manifold. It is shown that injectivity of the projective development map of the curve implies the injectivity of the immersion to Moebius spaces.