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

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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.