Geometric structure of complete Riemannian manifolds and the scalar curvature equation

完全黎曼流形的几何结构和标量曲率方程

• 批准号: 11640209
• 负责人: KATO Shin
• 金额: $ 2.3万
• 依托单位: Osaka City University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640209/
• 关键词: scalar curvature; conformal deformation; 共形変形

This project is a reserch on the scalar curvature equation that is an analytic formulation of the problem "Which kind of smooth function on a Riemannian manifold can be realized as the scalar curvature of a Riemannian metric which is pointwise conformal to the given metric ?" In this project, we deal with the case of noncompact complete Riemannian manifolds.To take a bload view of the problem, as a reserch of geometric structure of complete Riemannian manifolds, we held, in 1999-2002, a series of meetings on the variational problems, e.g. harmonic maps, spectral geometry and the collapse of Riemannian manifolds, the graph theory, the motion of elastic curves etc., each of which has something in common with the scalar curvature equation.In 2000, the head investigator wrote a survey on the scalar curvature equation on open Riemannian manifolds.In 2000-2002, he also investigated on the separating phenomenon which occurs with concentration of curvature, which we can regard as a kind of bubble, and got some estimates on the separation of a Riemannian manifold by a new invariant called relative weight of end-pairs, in the model case of minimal surfaces.

本课题研究的是“黎曼流形上哪一种光滑函数可以实现为黎曼度规点向保形的标量曲率”问题的解析形式的标量曲率方程。在这个项目中，我们处理非紧化完全黎曼流形的情况。从整体上看，作为完全黎曼流形几何结构的研究，我们于1999-2002年召开了一系列与标量曲率方程有共通之处的变分问题的会议，如调和映射、谱几何与黎曼流形的坍缩、图论、弹性曲线的运动等。2000年，首席研究员写了一篇关于开放黎曼流形上的标量曲率方程的综述。在2000-2002年，他还研究了曲率集中时发生的分离现象，我们可以把曲率看作一种气泡，并在最小曲面的模型情况下，用一个新的称为端对相对权的不变量对黎曼流形的分离给出了一些估计。

项目成果

Shin KATO: "The scalarlcurvature equation on open Riemannian manifolds (in Japanese)"Sugaku. 51. 225-240 (1999)

Shin KATO：“开黎曼流形上的标量曲率方程（日语）”Sugaku。

加藤 信: "η-end catenoidのend対のweightについて"Lecture Note Series in Mathematics, Osaka University. 7. 93-108 (2002)

加藤诚：“关于 η 端悬链线末端对的重量”，大阪大学数学讲义系列，7. 93-108 (2002)。

Shin KATO: "The Scalar Curvature Equation on Open Riemannicm Manifolds"SUGAKU. (未定).

Shin KATO：“开放黎曼流形上的标量曲率方程”SUGAKU（待定）。

Shin KATO, Masaaki UMEHARA, Kotaro YAMADA: "General existence of minimal surfaces of genes zero with catenoidal ends and prescribed flux"Comm. Anal. Geom.. 8. 83-114 (2000)

Shin KATO、Masaaki UMEHARA、Kotaro YAMADA：“具有链状末端和规定通量的基因零的最小表面的一般存在性”Comm。

Shin KATO: "The scalar curvature equation on open Riemannian manifolds"Sugaku Expositions. 14. 219-236 (2001)

Shin KATO：“开黎曼流形上的标量曲率方程”Sugaku Expositions。