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

这个博士项目是纯数学的，动机来自数学物理(数学广义相对论)，目的是证明两种类型的准局部质量：霍金质量和巴特尼克质量的变分和刚性性质。虽然本文将限于三维黎曼环境流形的情况，但这些概念是(数学)广义相对论的基本概念，因为它们对包含在类黎曼时空切片中的固体区域内的准局部质量进行建模。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PenRose不等式(Riemannian情形)的证明是基于逆平均曲率流下的Hawking质量的单调性的。这个项目的第一个目标是证明，如果对于3D黎曼流形中的每个点，存在一个小邻域，其中任何曲面的Hawking质量为零(实际上是非正的)，那么整个空间一定是平坦的。这可以被视为霍金质量的局部到全球的属性。然后，利用惠斯肯和伊尔马宁的观点，它将某些表面的霍金和巴特尼克质量联系起来。这使得对现有刚性定理的新证明和给出Bartnik质量的渐近下界的新结果成为可能。证明将在于微分几何、偏微分方程式和数学广义相对论的交集。非线性分析和偏微分方程组将发挥关键的技术作用来攻击霍金质量定理，因为最优(Willmore泛函的约束临界点)曲面将被发现和利用。我们将计算这些最优表面的霍金质量的一个新的泰勒展开式，然后需要一个标量曲率下界(对应于广义相对论中的主导能量条件)来得出刚性。有限周长集理论将被用来证明微扰测地球体(先前发现的最佳曲面)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.