Construction of Riemannian manifolds with scalar curvature constraints and applications to general relativity

这个项目涉及通过几何和分析技术构造具有标量曲率约束的黎曼流形，满足广义相对论中公开问题所激发的性质。更准确地说，宇宙中的孤立系统（如星星、星系或黑洞）可以建模为爱因斯坦方程的解，该方程构成了一组高度非线性的几何PDE。研究爱因斯坦方程解的一个非常成功的方法是通过其相关的柯西问题，其中宇宙的初始状态由黎曼流形表示，其初始速度由对称的2-张量表示，使得流形和2-张量满足所谓的约束方程，特别是对流形的标量曲率施加条件。 Choquet-Bruhat [1952]证明了这些约束足以保证局部解的存在性。不幸的是，求解约束方程是一项困难的任务，除了主要由Lichnerowicz和约克开发的保角方法外，没有多少方法可以这样做。因此，开发新的技术来解决这些问题，即构造黎曼流形与满足约束方程的对称2-张量是非常有意义的。最近，Racz [2016]提出了一种新的方法，其中约束方程可以重写为抛物-双曲系统，可以保证局部存在性。然而，不知道可以施加哪些条件来获得全局存在性和渐近平坦性（即，孤立系统模型）。对于对称2-张量恒为零的情形，Bartnik [1993]建立了这样的条件。对Bartnik构造的修改允许一个非平凡的对称2-张量来证明Racz系统的整体存在性;这将导致约束方程的渐近平坦解。估计这些解的ADM质量（总质量的概念），并验证它们构成了Penrose不等式猜想成立的流形族。在对称2-张量恒为零的条件下，研究了正能量定理、黎曼Penrose不等式和拟局部质量的稳定性。也就是说，开发技术来研究具有非负标量曲率的渐近平坦黎曼流形序列（通过某些PDE方法获得）的收敛性，相对于流形之间距离的不同概念，例如Sormani-Wenger的内在平坦距离。然后，利用这些技巧来研究流形族正质量定理的稳定性，作为主要目标的一部分。

This project deals with the construction of Riemannian manifolds with scalar curvature constraints via geometric and analytic techniques, satisfying properties motivated by open questions in general relativity. More precisely, an isolated system in the universe (as a star, galaxy or black hole) can be modeled as a solution of the Einstein equations, which constitute a highly non-linear set of geometric PDE's. A very successful way to study solutions to the Einstein equations is by means of its associated Cauchy problem, in which the initial state of the universe is represented by a Riemannian manifold and its initial velocity by a symmetric 2-tensor, such that the manifold and the 2-tensor satisfy the so-called constraint equations, which in particular impose conditions on the scalar curvature of the manifold. Choquet-Bruhat [1952] proved that these constraints are sufficient to guarantee existence of a local solution. Unfortunately, solving the constraint equations is a difficult task, and besides the conformal method developed mainly by Lichnerowicz and York, not many methods are available to do so. It is therefore of high interest to develop new techniques to solve them, that is, to construct Riemannian manifolds together with symmetric 2-tensors satisfying the constraint equations. Recently, Racz [2016] proposed a new approach in which the constraint equations can be rewritten as a parabolic-hyperbolic system for which local existence can be guaranteed. However, it is unknown which conditions could be imposed to obtain global existence and asymptotic flatness (i.e., models of isolated systems). For the case that the symmetric 2-tensor is identically zero, such conditions were established by Bartnik [1993].The objectives of this project can be divided into two groups:Main objective. The adaptation of Bartnik's construction to allow a non-trivial symmetric 2-tensor to show global existence of Racz's system; this would lead to asymptotically flat solutions of the constraint equations. Estimate the ADM mass (a notion of total mass) of these solutions and verify that they constitute a family of manifolds for which the Penrose inequality conjecture holds.Secondary objective. Restricting to the case when the symmetric 2-tensor is identically zero, study the stability of the positive energy theorem, the Riemannian Penrose inequality and some notions of quasi-local mass. That is, develop techniques to study the convergence of sequences of asymptotically flat Riemannian manifolds (obtained via certain PDE methods) with non-negative scalar curvature, with respect to different notions of distances between manifolds, for example, Sormani-Wenger's intrinsic flat distance. Then, use these techniques to study the stability of the positive mass theorem for the family of manifolds obtained as part of the main objective.