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

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天然工具。在这个项目中，我们考虑了广义相对论中与霍金质量相关的黎曼流形中的Willmore泛函的可能推广。我们研究在一定约束条件下使该泛函最小化的曲面或满足相关偏微分方程的曲面。主要重点是研究环境歧管的几何形状对这些表面几何形状的影响。当环境流形是爱因斯坦方程的初始数据集时，这种关系通常具有物理解释。例如，一个孤立的重力系统的质心可以用包围相同体积的所有表面中面积最小的表面来定义。这些都是我们研究Willmore函数的效果。申请人以前的工作表明，Willmore泛函与小表面环境空间的标量曲率以及大尺度孤立引力系统的质量和质心之间存在紧密联系。我们还考虑了Willmore函数的版本，它有望捕获线性动量。

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.