Searcharxiv⌕ Search

arXiv subjects

Jeffrey L. Jauregui

Publications and source records attributed to Jeffrey L. Jauregui.

18 recordsLinked to original sources
arXiv2026

Maximum capacity of Bartnik data and a generalization of static metrics

Inspired by R. Bartnik's mass minimization problem in general relativity, we investigate a dual problem of maximizing the capacity among asymptotically flat extensions (with nonnegative scalar curvature) of some fixed two-dimensional boundary data. Using the method of Lagrange multipliers on the constraint space of scalar-flat extensions, we derive the variational condition satisfied by a maximal capacity extension. The resulting equation is an inhomogeneous generalization of the well-known static equation, now coupled with the Baird--Eells stress-energy tensor for a harmonic function. We analyze these ``harmonic-static'' metrics in a local sense, proving they have constant scalar curvature and serve as critical points for a metric-dependent Dirichlet energy functional. We conclude with a number of open questions.

math.DG↗
arXiv2026

Semicontinuity of capacity under pointed intrinsic flat convergence

The concept of the capacity of a compact set in $\mathbb R^n$ generalizes readily to noncompact Riemannian manifolds and, with more substantial work, to metric spaces (where multiple natural definitions of capacity are possible). Motivated by analytic and geometric considerations, and in particular Jauregui's definition of capacity-volume mass and Jauregui and Lee's results on the lower semicontinuity of the ADM mass and Huisken's isoperimetric mass, we investigate how the capacity functional behaves when the background spaces vary. Specifically, we allow the background spaces to consist of a sequence of local integral current spaces converging in the pointed Sormani--Wenger intrinsic flat sense. For the case of volume-preserving ($\mathcal{VF}$) convergence, we prove two theorems that demonstrate an upper semicontinuity phenomenon for the capacity: one version is for balls of a fixed radius centered about converging points; the other is for Lipschitz sublevel sets. Our approach is motivated by Portegies' investigation of the semicontinuity of eigenvalues under $\mathcal{VF}$ convergence. We include examples to show the semicontinuity may be strict, and that the volume-preserving hypothesis is necessary. Finally, there is a discussion on how capacity and our results may be used towards understanding the general relativistic total mass in non-smooth settings.

math.DG↗
arXiv2020

Scalar curvature and the relative capacity of geodesic balls

In a Riemannian manifold, it is well known that the scalar curvature at a point can be recovered from the volumes (areas) of small geodesic balls (spheres). We show the scalar curvature is likewise determined by the relative capacities of concentric small geodesic balls. This result has motivation from general relativity (as a complement to a previous study by the author of the capacity of large balls in an asymptotically flat manifold) and from weak definitions of nonnegative scalar curvature. It also motivates a conjecture (inspired by the famous volume conjecture of Gray and Vanhecke), regarding whether Euclidean-like behavior of the relative capacity on the small scale is sufficient to characterize a space as flat.

math.DG↗
arXiv2020

ADM mass and the capacity-volume deficit at infinity

Based on the isoperimetric inequality, G. Huisken proposed a definition of total mass in general relativity that is equivalent to the ADM mass for (smooth) asymptotically flat 3-manifolds of nonnegative scalar curvature, but that is well-defined in greater generality. In a similar vein, we use the isocapacitary inequality (bounding capacity from below in terms of volume) to suggest a new definition of total mass. We prove an inequality between it and the ADM mass, and prove the reverse inequality with harmonically flat asymptotics, or, with general asymptotics, for exhaustions by balls (as opposed to arbitrary compact sets). This approach to mass may have applications to problems involving low regularity metrics and convergence in general relativity, and may have some advantages relative to the isoperimetric mass. Some conjectures, analogs of known results for CMC surfaces and isoperimetric regions, are proposed.

math.DG↗
arXiv2019

Smoothing the Bartnik boundary conditions and other results on Bartnik's quasi-local mass

Quite a number of distinct versions of Bartnik's definition of quasi-local mass appear in the literature, and it is not a priori clear that any of them produce the same value in general. In this paper we make progress on reconciling these definitions. The source of discrepancies is two-fold: the choice of boundary conditions (of which there are three variants) and the non-degeneracy or "no-horizon" condition (at least six variants). To address the boundary conditions, we show that given a 3-dimensional region $Ω$ of nonnegative scalar curvature ($R \geq 0$) extended in a Lipschitz fashion across $\partial Ω$ to an asymptotically flat 3-manifold with $R \geq 0$ (also holding distributionally along $\partial Ω$), there exists a smoothing, arbitrarily small in $C^0$ norm, such that $R \geq 0$ and the geometry of $Ω$ are preserved, and the ADM mass changes only by a small amount. With this we are able to show that the three boundary conditions yield equivalent Bartnik masses for two reasonable non-degeneracy conditions. We also discuss subtleties pertaining to the various non-degeneracy conditions and produce a nontrivial inequality between a no-horizon version of the Bartnik mass and Bray's replacement of this with the outward-minimizing condition.

math.DG↗
arXiv2019

Lower semicontinuity of ADM mass under intrinsic flat convergence

A natural question in mathematical general relativity is how the ADM mass behaves as a functional on the space of asymptotically flat 3-manifolds of nonnegative scalar curvature. In previous results, lower semicontinuity has been established by the first-named author for pointed $C^2$ convergence, and more generally by both authors for pointed $C^0$ convergence (all in the Cheeger--Gromov sense). In this paper, we show this behavior persists for the much weaker notion of pointed Sormani--Wenger intrinsic flat ($\mathcal{F}$) volume convergence, under natural hypotheses. We consider smooth manifolds converging to asymptotically flat local integral current spaces (a new definition), using Huisken's isoperimetric mass as a replacement for the ADM mass. Along the way we prove results of independent interest about convergence of subregions of $\mathcal{F}$-converging sequences of integral current spaces.

math.DG↗
arXiv2019

Embeddings, immersions and the Bartnik quasi-local mass conjectures

Given a Riemannian 3-ball $(\bar B, g)$ of non-negative scalar curvature, Bartnik conjectured that $(\bar B, g)$ admits an asymptotically flat (AF) extension (without horizons) of the least possible ADM mass, and that such a mass-minimizer is an AF solution to the static vacuum Einstein equations, uniquely determined by natural geometric conditions on the boundary data of $(\bar B, g)$. We prove the validity of the second statement, i.e.~such mass-minimizers, if they exist, are indeed AF solutions of the static vacuum equations. On the other hand, we prove that the first statement is not true in general; there is a rather large class of bodies $(\bar B, g)$ for which a minimal mass extension does not exist.

math.DG↗
arXiv2018

Penrose-type inequalities with a Euclidean background

The Riemannian Penrose inequality (RPI) bounds from below the ADM mass of asymptotically flat manifolds of nonnegative scalar curvature in terms of the total area of all outermost compact minimal surfaces. The general form of the RPI is currently known for manifolds of dimension up to seven. In the present work, we prove a Penrose-like inequality that is valid in all dimensions, for conformally flat manifolds. Our inequality treats the area contributions of the minimal surfaces in a more favorable way than the RPI, at the expense of using the smaller Euclidean area (rather than the intrinsic area). We give an example in which our estimate is sharper than the RPI when many minimal surfaces are present. We do not require the minimal surfaces to be outermost. We also generalize the technique to allow for metrics conformal to a scalar-flat (not necessarily Euclidean) background, and prove a Penrose-type inequality without an assumption on the sign of scalar curvature. Finally, we derive a new lower bound for the ADM mass of a conformally flat, asymptotically flat manifold containing any number of zero area singularities.

math.DG↗
arXiv2018

Lower semicontinuity of the ADM mass in dimensions two through seven

The semicontinuity phenomenon of the ADM mass under pointed (i.e., local) convergence of asymptotically flat metrics is of interest because of its connections to nonnegative scalar curvature, the positive mass theorem, and Bartnik's mass-minimization problem in general relativity. In this paper, we extend a previously known semicontinuity result in dimension three for $C^2$ pointed convergence to higher dimensions, up through seven, using recent work of S. McCormick and P. Miao (which itself builds on the Riemannian Penrose inequality of H. Bray and D. Lee). For a technical reason, we restrict to the case in which the limit space is asymptotically Schwarzschild. In a separate result, we show that semicontinuity holds under weighted, rather than pointed, $C^2$ convergence, in all dimensions $n \geq 3$, with a simpler proof independent of the positive mass theorem. Finally, we also address the two-dimensional case for pointed convergence, in which the asymptotic cone angle assumes the role of the ADM mass.

math.DG↗
arXiv2016

Lower semicontinuity of mass under $C^0$ convergence and Huisken's isoperimetric mass

Given a sequence of asymptotically flat 3-manifolds of nonnegative scalar curvature with outermost minimal boundary, converging in the pointed $C^0$ Cheeger--Gromov sense to an asymptotically flat limit space, we show that the total mass of the limit is bounded above by the liminf of the total masses of the sequence. In other words, total mass is lower semicontinuous under such convergence. In order to prove this, we use Huisken's isoperimetric mass concept, together with a modified weak mean curvature flow argument. We include a brief discussion of Huisken's work before explaining our extension of that work. The results are all specific to three dimensions.

math.DG↗
arXiv2014

On the lower semicontinuity of the ADM mass

The ADM mass, viewed as a functional on the space of asymptotically flat Riemannian metrics of nonnegative scalar curvature, fails to be continuous for many natural topologies. In this paper we prove that lower semicontinuity holds in natural settings: first, for pointed Cheeger--Gromov convergence (without any symmetry assumptions) for $n=3$, and second, assuming rotational symmetry, for weak convergence of the associated canonical embeddings into Euclidean space, for $n \geq 3$. We also apply recent results of LeFloch and Sormani to deal with the rotationally symmetric case, with respect to a pointed type of intrinsic flat convergence. We provide several examples, one of which demonstrates that the positive mass theorem is implied by a statement of the lower semicontinuity of the ADM mass.

math.DG↗
arXiv2014

Time flat surfaces and the monotonicity of the spacetime Hawking mass II

In this sequel paper we give a shorter, second proof of the monotonicity of the Hawking mass for time flat surfaces under spacelike uniformly area expanding flows in spacetimes that satisfy the dominant energy condition. We also include a third proof which builds on a known formula and describe a class of sufficient conditions of divergence type for the monotonicity of the Hawking mass. These flows of surfaces may have connections to the problem in general relativity of bounding the total mass of a spacetime from below by the quasi-local mass of spacelike 2-surfaces in the spacetime.

math.DG↗
arXiv2014

On curves with nonnegative torsion

We provide new results and new proofs of results about the torsion of curves in $\mathbb{R}^3$. Let $γ$ be a smooth curve in $\mathbb{R}^3$ that is the graph over a simple closed curve in $\mathbb{R}^2$ with positive curvature. We give a new proof that if $γ$ has nonnegative (or nonpositive) torsion, then $γ$ has zero torsion and hence lies in a plane. Additionally, we prove the new result that a simple closed plane curve, without any assumption on its curvature, cannot be perturbed to a closed space curve of constant nonzero torsion. We also prove similar statements for curves in Lorentzian $\mathbb{R}^{2,1}$ which are related to important open questions about time flat surfaces in spacetimes and mass in general relativity.

math.DG↗
arXiv2013

Time flat surfaces and the monotonicity of the spacetime Hawking mass

We identify a condition on spacelike 2-surfaces in a spacetime that is relevant to understanding the concept of mass in general relativity. We prove a formula for the variation of the spacetime Hawking mass under a uniformly area expanding flow and show that it is nonnegative for these so-called "time flat surfaces." Such flows generalize inverse mean curvature flow, which was used by Huisken and Ilmanen to prove the Riemannian Penrose inequality for one black hole. A flow of time flat surfaces may have connections to the problem in general relativity of bounding the mass of a spacetime from below by the quasi-local mass of a spacelike 2-surface contained therein.

math.DG↗
arXiv2013

Fill-ins of nonnegative scalar curvature, static metrics, and quasi-local mass

Consider a triple of "Bartnik data" $(Σ, γ,H)$, where $Σ$ is a topological 2-sphere with Riemannian metric $γ$ and positive function $H$. We view Bartnik data as a boundary condition for the problem of finding a compact Riemannian 3-manifold $(Ω,g)$ of nonnegative scalar curvature whose boundary is isometric to $(Σ,γ)$ with mean curvature $H$. Considering the perturbed data $(Σ, γ, λH)$ for a positive real parameter $λ$, we find that such a "fill-in" $(Ω,g)$ must exist for $λ$ small and cannot exist for $λ$ large; moreover, we prove there exists an intermediate threshold value. The main application is the construction of a new quasi-local mass, a concept of interest in general relativity. This mass has the nonnegativity property, but differs from many other definitions in that it tends to vanish on static vacuum (as opposed to flat) regions. We also recognize this mass as a special case of a type of twisted product of quasi-local mass functionals. Several ideas in this paper draw on work of Bray, Brendle--Marques--Neves, Corvino, Miao, and Shi--Tam.

math.DG↗
arXiv2012

A geometric theory of zero area singularities in general relativity

The Schwarzschild spacetime metric of negative mass is well-known to contain a naked singularity. In a spacelike slice, this singularity of the metric is characterized by the property that nearby surfaces have arbitrarily small area. We develop a theory of such "zero area singularities" in Riemannian manifolds, generalizing far beyond the Schwarzschild case (for example, allowing the singularities to have nontrivial topology). We also define the mass of such singularities. The main result of this paper is a lower bound on the ADM mass of an asymptotically flat manifold of nonnegative scalar curvature in terms of the masses of its singularities, assuming a certain conjecture in conformal geometry. The proof relies on the Riemannian Penrose Inequality. Equality is attained in the inequality by the Schwarzschild metric of negative mass. An immediate corollary is a version of the Positive Mass Theorem that allows for certain types of incomplete metrics.

math.DG↗
arXiv2010

Invariants of the harmonic conformal class of an asymptotically flat manifold

Consider an asymptotically flat Riemannian manifold $(M,g)$ of dimension $n \geq 3$ with nonempty compact boundary. We recall the harmonic conformal class $[g]_h$ of the metric, which consists of all conformal rescalings given by a harmonic function raised to an appropriate power. The geometric significance is that every metric in $[g]_h$ has the same pointwise sign of scalar curvature. For this reason, the harmonic conformal class appears in the study of general relativity, where scalar curvature is related to energy density. Our purpose is to introduce and study invariants of the harmonic conformal class. These invariants are closely related to constrained geometric optimization problems involving hypersurface area-minimizers and the ADM mass. In the final section, we discuss possible applications of the invariants and their relationship with zero area singularities and the positive mass theorem.

math.DG↗