arXiv subjects

Rodrigo Avalos

Publications and source records attributed to Rodrigo Avalos.

13 recordsLinked to original sources

arXiv2025

On energy and its positivity in spacetimes with an expanding flat de Sitter background

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.

gr-qc↗

arXiv2025

Regularity of conformal structures on closed 3-manifolds

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.

math.DG↗

arXiv2025

Conformal Green functions and Yamabe metrics of Sobolev regularity

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 $q>\tfrac{n}{2}$ and should be of independent interest.

math.AP↗

arXiv2024

Sobolev regularity of compactified 3-manifolds and the ADM Center of Mass

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 question 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 shall first establish regularity properties of $L^{q'}$-solutions associated to the operator $\Delta_{\hat{g}}:W^{k,p}(S_2\hat{M})\to W^{k-2,p}(S_2\hat{M})$, for $k=1,2$ and $1 \frac{n}{2}$ and $S_2\hat{M}$ denotes the bundle of symmetric $(0,2)$-tensor fields. Appealing to these results, we establish Sobolev regularity 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 will allow us to address a version of a conjecture posed by C. Cederbaum and A. Sakovich concerning the convergence of the COM of such manifolds.

math.DG↗

arXiv2022

On the Rigidity of Cosmological Space-times

In this paper we analyse a family of geometrically well-behaved cosmological space-times $(V^{n+1},g)$, which are foliated by intrinsically isotropic space-like hypersurfaces $\{M_t\}_{t\in \mathbb{R}}$, which are orthogonal to a family of co-moving observers defined by a global time-like vector field $U$. In particular, this implies such space-times satisfy several of the well-known criteria for isotropic cosmological space-times, although, in the family in question, the simultaneity-spaces $(M_t,g_t)$ associated to $U$ can have as sectional curvature a sign-changing function $k(t)$. Being this clearly impossible in the FLRW family of standard cosmological space-times, it motivates us to revisit the geometric rigidity consequences of different definitions of isotropy available in the literature. In this analysis, we divide such definition according to whether the isometries involved are taken to be (local) space-time (STI space-times) or (local) space isometries (SI space-times) of $(M_t,g_t)$ for each $t$. This subtlety will be shown to be critical, proving that only when space-time isometries are considered one obtains the well-known rigidity properties associated with isotropic cosmological space-times. In particular SI space-times will be shown to be a strictly larger class than the STI ones, allowing a family of basic cosmological curvature change models which are not even locally isometric to any FLRW space-time.

gr-qc↗

arXiv2022

Rigidity Theorems for Asymptotically Euclidean $Q$-singular Spaces

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.

math.DG↗

arXiv2022

Einstein Type Systems on Complete Manifolds

In this paper, we study the coupled Einstein constraint equations on complete manifolds through the conformal method, focusing on non-compact manifolds with flexible asymptotics. This is physically well-motivated by standard cosmological space-times with non-compact Cauchy hypersurfaces, which favour general bounded geometry manifolds rather than a specific model for infinity. First, we prove an existence criterion on complete manifolds with appropriate barrier functions for physically well-motivated coupled systems. Then, in the bounded geometry case, we build barrier functions and thus show existence. We also prove an existence result on compact manifolds with boundary for a wider family of coupled systems.

math.AP↗

arXiv2021

On the Positive Energy Theorem for Stationary Solutions to Fourth-Order Gravity

In this paper we prove a positive energy theorem related to fourth-order gravitational theories, which is a higher-order analogue of the classical ADM positive energy theorem of general relativity. We will also show that, in parallel to the corresponding situation in general relativity, this result intersects several important problems in geometric analysis. For instance, it underlies positive mass theorems associated to the Paneitz operator, playing a similar role in the positive $Q$-curvature conformal prescription problem as the Schoen-Yau positive energy theorem does for the Yamabe problem. Several other links to $Q$-curvature analysis and rigidity phenomena are established.

math.DG↗

arXiv2021

Energy in Fourth Order Gravity

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.

math.DG↗

arXiv2019

Einstein-type elliptic systems

In this paper we analyse semi-linear systems of partial differential equations which are motivated by the conformal formulation of the Einstein constraint equations coupled with realistic physical fields on asymptotically Euclidean (AE) manifolds. In particular, electromagnetic fields give rise to this kind of system. In this context, under suitable conditions, we prove a general existence theorem for such systems, and, in particular, under smallness assumptions on the free parameters of the problem, we prove existence of far from CMC (near CMC) Yamabe positive (Yamabe non-positive) solutions for charged dust coupled to the Einstein equations, satisfying a trapped surface condition on the boundary. As a bypass, we prove a Helmholtz decomposition on AE manifolds with boundary, which extends and clarifies previously known results.

gr-qc↗

arXiv2019

Reduced thin-sandwich equations on manifolds euclidean at infinity and on closed manifolds: existence and multiplicity

The reduced thin-sandwich equations (RTSE) appear within Wheeler's thin-sandwich approach towards the Einstein constraint equations (ECE) of general relativity. It is known that these equations cannot be well-posed in general, but, on closed manifolds, sufficient conditions for well-posedness have been established. In particular, it has been shown that the RTSE are well posed in a neighbourhood of umbilical solutions of the constraint equations without conformal Killing fields. In this paper we will analyse such set of equations on manifolds euclidean at infinity in a neighbourhood of asymptotically euclidean (AE) solutions of the ECE. The main conclusion in this direction is that on AE-manifolds admitting a Yamabe positive metric, the solutions of the RTSE parametrize an open subset in the space of solutions of the ECE. Also, we show that in the case of closed manifolds, these equations are well-posed around umbilical solutions of the ECE admitting Killing fields and present some relevant examples. Finally, it will be shown that in the set of umbilical solutions of the vacuum ECE on closed manifolds, the RTSE are generically well-posed.

gr-qc↗

arXiv2017

On the $C^k$-embedding of Lorentzian manifolds in Ricci-flat spaces

In this paper we investigate the problem of non-analytic embeddings of Lorentzian manifolds in Ricci-flat semi-Riemannian spaces. In order to do this, we first review some relevant results in the area, and then motivate both the mathematical and physical interest in this problem. We show that any $n$-dimensional compact Lorentzian manifold $(M^{n},g)$, with $g$ in the Sobolev space $H_{s+3}$, $s>\frac{n}{2}$, admits an isometric embedding in an $(2n+2)$-dimensional Ricci-flat semi-Riemannian manifold. The sharpest result available for this type of embeddings, in the general setting, comes as a corollary of Greene's remarkable embedding theorems [R. Greene, Mem. Am. Math. Soc. 97, 1 (1970)], which guarantee the embedding of a compact $n$-dimensional semi-Riemannian manifold into an $n(n+5)$-dimensional semi-Euclidean space, thereby guaranteeing the embedding into a Ricci-flat space with the same dimension. The theorem presented here improves this corollary in $n^{2}+3n-2$ codimensions by replacing the Riemann-flat condition with the Ricci-flat one from the beginning. Finally, we will present a corollary of this theorem, which shows that a compact strip in an $n$-dimensional globally hyperbolic space-time can be embedded in a $(2n+2)$-dimensional Ricci-flat semi-Riemannian manifold.

gr-qc↗

arXiv2014

Perihelion advances for the orbits of Mercury, Earth and Pluto from Extended Theory of General Relativity (ETGR)

We explore the geodesic movement on an effective four-dimensional hypersurface that is embedded in a five-dimensional Ricci-flat manifold described by a canonical metric, in order to applying to planetary orbits in our solar system. Some important solutions are given, which provide the standard solutions of general relativity without any extra force component. We study the perihelion advances of Mercury, the Earth and Pluto using the extended theory of general relativity (ETGR). Our results are in very good agreement with observations and show how the foliation is determinant to the value of the perihelion's advances. Possible applications are not limited to these kinds of orbits.

physics.space-ph↗