Gemetric Structures on Manifolds and Global Analysis

在流形的许多几何结构中，我们主要对那些与共形几何密切相关的结构感兴趣。以下是本研究项目的一些主要成果：标量曲率方程。这个方程描述了黎曼度规的共形变化下的标量曲率。对非紧流形作了系统的分析，使具有规定标量曲率的完全共形度量空间更加清晰。Weyl结构。这是一个与给定保角类相容的无扭转仿射连接。证明了Ricci曲率是Weyl结构的完全不变量。此外，还对紧流形上的共形平坦Einstein-Weyl结构进行了分类。莫比乌斯几何学。给定拓扑类型的球面上的正则闭曲线，当曲线最多有5个自交点时，其最小顶点数完全确定。我们还引入了正则曲线的Schwarzian导数。这导致了Nehari型注入性结果的新证据。一个要点是流形的共形结构推导出流形上正则曲线的可积投影结构。结果表明，曲线投影发展图的注入率反映了对莫比乌斯空间的浸没注入率。

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.