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The positive energy theorems are a fundamental pillar in mathematical general relativity. Originally proved by Schoen-Yau and later Witten, these theorems were established for asymptotically flat manifolds where the metric tends to the standard Euclidean metric and whose second fundamental form decays to zero at infinity. This ansatz on the metric and second fundamental form is motivated by the desire to model an isolated gravitational system with a Minkowski space background for the spacetime. However, actual astrophysical massive objects are not truly isolated but rather exist within an expanding cosmological universe, where the second fundamental form is umbilic. With this in mind, we seek a notion of energy for initial data sets with an umbilic second fundamental form. In this work, we present a definition of energy in such an expanding cosmological setting. Instead of Minkowski space, we take de Sitter space as the background spacetime, which, when written in flat-expanding coordinates, is foliated by umbilic hypersurfaces each isometric to Euclidean 3-space. This cosmological setting necessitates a quasi-local energy definition, as the presence of a cosmological horizon in de Sitter space obstructs a global one. We define energy in this quasi-local setting by adapting the Liu-Yau energy to our framework and establish positivity of this energy for certain bounded values of the cosmological constant.

Related project(s):
41Geometrically defined asymptotic coordinates in general relativity

In this paper we prove some rigidity theorems associated to \(Q\)-curvature analysis on asymptotically Euclidean (AE) manifolds, which are inspired by the analysis of conservation principles within fourth order gravitational theories. A central object in this analysis is a notion of fourth order energy, previously analysed by the authors, which is subject to a positive energy theorem. We show that this energy can be more geometrically rewritten in terms of a fourth order analogue to the Ricci tensor, which we denote by \(J_g\). This allows us to prove that Yamabe positive \(J\)-flat AE manifolds must be isometric to Euclidean space. As a by product, we prove that this \(J\)-tensor provides a geometric control for the optimal decay rates at infinity. This last result reinforces the analogy of \(J\) as a fourth order analogue to the Ricci tensor.

Related project(s):
41Geometrically defined asymptotic coordinates in general relativity

We provide a full resolution of the Yamabe problem on closed 3-manifolds for Riemannian metrics of Sobolev class \(W^{2,q}\) with \(q>3\). This requires developing an elliptic theory for the conformal Laplacian for rough metrics and establishing existence, regularity and a delicate blow-up analysis for its Green function. Most of the analytical work is carried out in dimensions \(n\geq 3\) and for \(W^{2,q}\)-Riemannian metrics with \(q>\frac{n}{2}\) and should be of independent interest.

Related project(s):
41Geometrically defined asymptotic coordinates in general relativity

It is well known in Riemannian geometry that the metric components have the best regularity in harmonic coordinates. These can be used to characterize the most regular element in the isometry class of a rough Riemannian metric. In this work, we study the conformal analogue problem on closed 3-manifolds: given a Riemannian metric \(g\) of class \(W^{2,q}\)  with \(q>3\), we characterize when a more regular representative exists in its conformal class. We highlight a deep link to the Yamabe problem for rough metrics and present some immediate applications to conformally flat, static and Einstein manifolds

Related project(s):
41Geometrically defined asymptotic coordinates in general relativity

In this paper we make a detailed analysis of conservation principles in the context of a family of fourth-order gravitational theories generated via a quadratic Lagrangian. In particular, we focus on the associated notion of energy and start a program related to its study. We also exhibit examples of solutions which provide intuitions about this notion of energy which allows us to interpret it, and introduce several study cases where its analysis seems tractable. Finally, positive energy theorems are presented in restricted situations.

Related project(s):
41Geometrically defined asymptotic coordinates in general relativity

In this paper we address the existence of preferred asymptotic coordinates on asymptotically Euclidean (AE) manifolds \((M^3,g)\) such that \(g\) admits an asymptotically Schwarzschildian first order expansion, based purely on a priori geometric conditions, which will then be used to establish geometric criteria guaranteeing the convergence of the ADM center of mass (COM). This problem is analysed relating it to the study of the regularity of conformal compactifications of such manifolds, which is itself explored via elliptic theory for operators with coefficients of very limited regularity. With these related problems in mind, we first establish Sobolev regularity properties of conformally compactified AE 3-manifolds via the decay of the Cotton tensor, improving on previous results. This allows us to construct preferred asymptotic coordinates on such AE manifolds where the metric has a first order Schwarzschildian expansion, which in turn allows us to address a version of a conjecture posed by C. Cederbaum and A. Sakovich concerning the convergence of the COM of such manifolds.

Related project(s):
41Geometrically defined asymptotic coordinates in general relativity