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

本项目通过几何和解析技术处理具有标量曲率约束的黎曼流形的构造，满足广义相对论中开放问题的性质。更准确地说，宇宙中的一个孤立系统（如恒星、星系或黑洞）可以被建模为爱因斯坦方程的一个解，它构成了一组高度非线性的几何偏微分方程。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，这样流形和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艰巨的任务，除了主要由Lichnerowicz和York开发的保形方法外，没有多少方法可以做到这一点。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。最近，Racz[2016]提出了一种新的方法，该方法将约束方程改写为可以保证局部存在的抛物-双曲系统。然而，目前尚不清楚可以施加哪些条件来获得全局存在性和渐近平坦性（即孤立系统的模型）。对于对称2张量同零的情况，Bartnik[1993]建立了这样的条件。本项目的目标可分为两组：主要目标。对Bartnik构造的改进使得一个非平凡对称2张量可以证明Racz系统的整体存在性这将导致约束方程的渐近平解。估计这些解的ADM质量（总质量的概念），并验证它们构成了彭罗斯不等式猜想成立的流形族。次要目标。在对称2张量等于零的情况下，研究了正能量定理、黎曼彭罗斯不等式和准局部质量的一些概念的稳定性。也就是说，发展研究具有非负标量曲率的渐近平坦黎曼流形序列（通过一定的PDE方法得到）在不同流形间距离概念下的收敛性的技术，例如Sormani-Wenger的固有平坦距离。然后，利用这些技术来研究得到的流形族的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.