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

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.

该项目介绍了通过几何和分析技术具有标态曲率约束的黎曼流形的构建，满足了由一般相对论中的开放性问题激励的属性。更确切地说，可以将宇宙中的孤立系统（作为恒星，星系或黑洞）建模为爱因斯坦方程的解决方案，该解决方案构成了高度非线性的几何PDE。一种非常成功的研究爱因斯坦方程解决方案的方法是通过其相关的库奇问题，其中宇宙的初始状态由riemannian歧管及其初始速度来表示，以对称的2张量，使得歧管和2张量子的典型构造方程式在典型的约束方程中，这在典型的构造方程中，这是对典型的条件的陈述。 Choquet-Bruhat [1952]证明，这些约束足以保证存在局部解决方案。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开发的共形方法外，没有多少方法可用。因此，开发新技术来解决它们是很高的兴趣，也就是说，构建Riemannian流形以及满足约束方程的对称的2张量。最近，Racz [2016]提出了一种新方法，其中可以将约束方程式重写为抛物线式杂种系统，可以保证该系统的局部存在。但是，尚不清楚哪些条件可以实现获得全球存在和渐近平坦度（即孤立系统的模型）。对于对称2 tensor的情况相同的情况，Bartnik [1993]建立了此类条件。该项目的目标可以分为两组：主要目标。 Bartnik的结构适应允许非平凡的对称2 tensor显示RACZ系统的全球存在；这将导致约束方程的渐近平面解决方案。估计这些解决方案的ADM质量（总质量的概念），并验证它们构成了penrose不平等猜想所构成的流形家族。限制了对称2量量相同的零，研究正能定理的稳定性，riemannian penrose不平等和一些准局部质量的概念。也就是说，开发技术来研究具有非负标量曲率的渐近平坦的riemannian歧管序列（通过某些PDE方法获得），相对于歧管之间的不同距离概念，例如，sormani-wengengengengengengengengani-wengenger的固有平整距离。然后，使用这些技术来研究作为主要目标一部分获得的流形家族的正质量定理的稳定性。