Quasi-local mass and 3D Riemannian manifolds with curvature lower bounds

这个博士项目是纯数学，动机来自数学物理（数学广义相对论），旨在证明两种类型的准局部质量的变分和刚性性质：霍金质量和Bartnik质量。虽然论文将限制在三维的情况下，黎曼环境流形，这些概念是基本的，（数学）广义相对论，因为他们模拟了时空的黎曼类空切片中包含在固体区域内的准局部质量。霍金质量一直是现代数学广义相对论的关键技术工具，例如，Huisken和Ilmanen对著名的彭罗斯不等式的证明（黎曼情况）是基于霍金质量在反平均曲率流下的单调性。这个项目的第一个目标是证明，如果对3D黎曼流形中的每一点都存在一个小邻域内的任何表面的霍金质量为零（实际上非正），那么整个空间必须是平坦的。这可以被看作是霍金质量的局部到全局性质。然后，使用Huisken和Ilmanen的想法，它将某些表面的霍金和Bartnik质量联系起来。这使得一个新的证明现有的刚性定理和一个新的结果给出一个渐近下界，无论是Bartnik质量。 证明将在于微分几何，偏微分方程和数学广义相对论的交叉点。非线性分析和偏微分方程将在攻击霍金质量定理方面发挥关键的技术作用，因为最佳（Willmore泛函的约束临界点）表面将被发现和利用。将计算这些最佳表面的霍金质量的新泰勒展开，然后需要标量曲率下限（对应于广义相对论中的主导能量条件）来得出刚性。有限周长集的理论将被用来证明扰动测地球面（之前发现的最优曲面）的关键性质，结合霍金质量的展开，将证明巴特尼克质量的结果。

This PhD project is in pure mathematics, with motivations coming from mathematical physics (mathematical general relativity) and aims to prove variational and rigidity properties of two types of quasi-local mass: Hawking mass and Bartnik mass. Though the thesis will restrict to the case of a three-dimensional, Riemannian ambient manifold, these concepts are fundamental in (mathematical) general relativity because they model the quasi-local mass contained inside a solid domain in a Riemannian space-like slice of a space-time.The Hawking mass has been a key technical tool in modern mathematical general relativity, for instance the proof by Huisken and Ilmanen of the celebrated Penrose inequality (Riemannian case) is based in the monotonicity of the Hawking mass under inverse mean curvature flow.The first goal of this project is to show that if for every point in a 3D Riemannian manifold there exists a small neighbourhood inside which the Hawking mass of any surface is zero (in fact non-positive), then the whole space must be flat. This can be seen as a local-to-global property of the Hawking mass. Then, using ideas from Huisken and Ilmanen, it relates the Hawking and Bartnik masses of certain surfaces. This enables a new proof of an existing rigidity theorem and a new result giving an asymptotic lower bound, both for the Bartnik mass. The proofs will lie in the intersection of differential geometry, partial differential equations, and mathematical general relativity. Non-linear analysis and partial differential equations will play a key technical role to attack the Hawking mass theorem, because optimal (constrained critical points of the Willmore functional) surfaces will be found and exploited. A new Taylor expansion of the Hawking mass of these optimal surfaces will be computed, and a scalar curvature lower bound (corresponding to the Dominant Energy condition in general relativity) will then be necessary to conclude the rigidity. The theory of sets of finite perimeter will be used to prove key properties of perturbed geodesic spheres (the optimal surfaces found earlier), which, combined with the expansion of the Hawking mass, will prove the Bartnik mass results.