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Geometry of Willmore surfaces in Riemannian manifolds and applications to General Relativity

黎曼流形中威尔莫尔曲面的几何及其在广义相对论中的应用

基本信息

批准号：
200100693
负责人：
Professor Dr. Jan Metzger
金额：
--
依托单位：
Institut für Mathematik
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2011
资助国家：
德国
起止时间：
2010-12-31 至 2013-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/200100693?language=en
关键词：
Geometry Willmore surfaces Riemannian manifolds

项目摘要

Geometric variational problems and related partial differential equations are natural tools in order to study the geometry of Riemannian manifolds and to define physical quantities in the theory of general relativity. In this project we consider possible generalizations of the Willmore functional in Riemannian manifolds related to the Hawking mass in general relativity. We study surfaces that minimize this functional subject to certain constraints or surfaces satisfying related partial differential equations. The main focus is to investigate the influence of the geometry of the ambient manifold to the geometry of these surfaces. Often such relations have a physical interpretation when the ambient manifold is an initial data set for the Einstein equations. For example, the center of mass of an isolated gravitating system can be defined using surfaces minimizing area among all surfaces enclosing the same volume. These are the kind of effects we study for the Willmore functional. Previous work of the applicant suggests that there is a tight connection of the Willmore functional to the scalar curvature of the ambient space for small surfaces and to the mass and the center of mass of an isolated gravitating system in the large scale. We also consider versions of the Willmore functional which are expected to capture the linear momentum.

几何变分问题和相关的偏微分方程是研究黎曼流形几何和定义广义相对论中物理量的自然工具。在这个项目中，我们考虑了广义相对论中与霍金质量有关的黎曼流形中的Willmore泛函的可能的推广。我们研究表面，尽量减少这个功能受到一定的约束或表面满足相关的偏微分方程。主要的重点是调查的几何形状的环境流形的几何形状的影响，这些表面。当环境流形是爱因斯坦方程的初始数据集时，这种关系往往有物理解释。例如，一个孤立的引力系统的质心可以用最小化包围相同体积的所有表面中的面积的表面来定义。这些就是我们研究的Willmore泛函的影响。申请人的先前工作表明，对于小表面，Willmore泛函与周围空间的标量曲率以及与大尺度中的孤立引力系统的质量和质心存在紧密联系。我们还考虑版本的Willmore功能，预计将捕获的线性动量。