Searcharxiv⌕ Search

arXiv subjects

Gregory J. Galloway

Publications and source records attributed to Gregory J. Galloway.

At least 19 recordsLinked to original sources
arXiv2026

Failure of local equi-Lipschitzness for families of Lorentz distances to Cauchy surface foliations

The families of Lorentzian distance functions to and from points along a complete timelike line in a spacetime are known to be locally equi-Lipschitz continuous in a neighborhood of the line. This is an essential component in the proof of the classical Lorentzian splitting theorems. We show that this property fails in general for families of Lorentzian distances to and from the level sets of a given Cauchy temporal function. Moreover, we formulate conjectures based on the existence of Cauchy temporal functions in cosmological and timelike geodesically complete spacetimes such that the Lorentz distances to its level sets are equi-Lipschitz, which are equivalent to Bartnik's splitting conjecture.

math.DG↗
arXiv2026

A CMC existence result for expanding cosmological spacetimes

We establish a new CMC (constant mean curvature) existence result for cosmological spacetimes, i.e., globally hyperbolic spacetimes with compact Cauchy surfaces satisfying the strong energy condition. If the spacetime contains an expanding Cauchy surface and is future timelike geodesically complete, then the spacetime contains a CMC Cauchy surface. This result settles, under certain circumstances, a conjecture of the authors and a conjecture of Dilts and Holst. Our proof relies on the construction of barriers in the support sense, and the CMC Cauchy surface is found as the asymptotic limit of mean curvature flow. Analogous results are also obtained in the case of a positive cosmological constant $Λ> 0$. Lastly, we include some comments concerning the future causal boundary for cosmological spacetimes which pertain to the CMC conjecture of the authors.

math.DG↗
arXiv2026

Positive mass theorems for manifolds with ALH toroidal ends

In work with P. Chruściel, L. Nguyen and T.-T. Paetz [8], a positive mass theorem was obtained for asymptotically locally hyperbolic manifolds with boundary, having a toroidal end. The proof made use of properties of marginally outer trapped surfaces (MOTS). Here we present some new PMT results for such manifolds, but without boundary, which allow for other more general ends. The proofs, while still MOTS-based, involve a more elaborate technique (related to $μ$-bubbles) introduced in work of D. A. Lee, M. Lesourd, and R. Unger [20] for manifolds with an asymptotically flat end, and further developed in [23] for manifolds with an asymptotically hyperbolic end.

math.DG↗
arXiv2025

A simplified proof of a cosmological singularity theorem

In a previous paper [9], we proved the following singularity theorem applicable to cosmological models with a positive cosmological constant: if a four-dimensional spacetime satisfying the null energy condition contains a compact Cauchy surface which is expanding in all directions, then the spacetime is past null geodesically incomplete unless the Cauchy surface is topologically a spherical space. The proof in [9] made use of the positive resolution of the surface subgroup conjecture [15]. In this note, we demonstrate how the less-broadly-known positive resolution of the virtual positive first Betti number conjecture [1] provides a more streamlined and unified approach to the proof. We illustrate the theorem with some examples and analyze its rigidity under null geodesic completeness.

gr-qc↗
arXiv2025

Stability and rigidity of axisymmetric marginally outer trapped two-spheres

In [7], H. Bray, S. Brendle, and A. Neves studied rigidity properties of area-minimizing two-spheres in Riemannian three-manifolds with uniformly positive scalar curvature. In [13], these results were extended to marginally outer trapped surfaces (MOTS) in general initial data sets $(M^3,g,K)$ under a natural energy condition. In the present work, we refine the latter results to the setting of axisymmetric MOTS in initial data sets admitting a nontrivial Killing vector field. Conditions for the stability of such MOTS, as well as a new foliation lemma by axisymmetric surfaces of constant outward null expansion, are obtained. Finally, we discuss some aspects of the rotating Nariai spacetimes and their relation to these results.

math.DG↗
arXiv2025

A note on the Lorentzian splitting theorem

We present a version of the Lorentzian splitting theorem under a weakened Ricci curvature condition. The proof makes use of basic properties of achronal limits [19], [20], together with the geometric maximum principle for $C^0$ spacelike hypersurfaces in [1]. Our version strengthens a related result in [29] in the globally hyperbolic setting by removing a certain boundedness condition on the Ricci curvature.

math.DG↗
arXiv2025

Mean Curvature, Singularities and Time Functions in Cosmology

In this contribution, we study spacetimes of cosmological interest, without making any symmetry assumptions. We prove a rigid Hawking singularity theorem for positive cosmological constant, which sharpens known results. In particular, it implies that any spacetime with $\operatorname{Ric} \geq -ng$ in timelike directions and containing a compact Cauchy hypersurface with mean curvature $H \geq n$ is timelike incomplete. We also study the properties of cosmological time and volume functions, addressing questions such as: When do they satisfy the regularity condition? When are the level sets Cauchy hypersurfaces? What can one say about the mean curvature of the level sets? This naturally leads to consideration of Hawking type singularity theorems for Cauchy surfaces satisfying mean curvature inequalities in a certain weak sense.

gr-qc↗
arXiv2025

Rigidity aspects of Penrose's singularity theorem

In this paper, we study rigidity aspects of Penrose's singularity theorem. Specifically, we aim to answer the following question: if a spacetime satisfies the hypotheses of Penrose's singularity theorem except with weakly trapped surfaces instead of trapped surfaces, then what can be said about the global spacetime structure if the spacetime is null geodesically complete? In this setting, we show that we obtain a foliation of MOTS which generate totally geodesic null hypersurfaces. Depending on our starting assumptions, we obtain either local or global rigidity results. We apply our arguments to cosmological spacetimes (i.e., spacetimes with compact Cauchy surfaces) and scenarios involving topological censorship.

gr-qc↗
arXiv2024

Aspects of the geometry and topology of expanding horizons

The aim of this paper is to extend some basic results about marginally outer trapped surfaces to the context of surfaces having general null expansion. Motivated in part by recent work of Chai-Wan, we introduce the notion of $\mathfrak{g}$-stability for a general closed hypersurface $Σ$ in an ambient initial data set and prove that, under natural energy conditions, $Σ$ has positive Yamabe type, that is, $Σ$ admits a metric of positive scalar curvature, provided $Σ$ is $\mathfrak{g}$-stable. A similar result is obtained when $Σ$ is embedded in a null hypersurface of a spacetime satisfying the dominant energy condition. Area bounds under similar conditions are obtained in the case where $Σ$ is $2$-dimensional. Conditions implying $\mathfrak{g}$-stability are also discussed. Finally, we obtain a spacetime positive mass theorem for initial data sets with compact boundary $Σ$ of positive null expansion, assuming that the dominant energy condition is sufficiently strict near $Σ$. This extends recent results of Galloway-Lee and Lee-Lesourd-Unger.

gr-qc↗
arXiv2024

Some rigidity results for charged initial data sets

In this note, we consider some initial data rigidity results concerning marginally outer trapped surfaces (MOTS). As is well known, MOTS play an important role in the theory of black holes and, at the same time, are interesting spacetime analogues of minimal surfaces in Riemannian geometry. The main results presented here expand upon earlier works by the authors, specifically addressing initial data sets incorporating charge.

math.DG↗
arXiv2023

Remarks on the size of apparent horizons

Marginally outer trapped surfaces (also referred to as apparent horizons) that are stable in 3-dimensional initial data sets obeying the dominant energy condition strictly are known to satisfy an area bound. The main purpose of this note is to show (in several ways) that such surfaces also satisfy a diameter bound.

gr-qc↗
arXiv2023

A note on null distance and causality encoding

Under natural conditions, the null distance introduced by Sormani and Vega [10] is a metric space distance function on spacetime, which, in a certain precise sense, can encode the causality of spacetime. The null distance function requires the choice of a time function. The purpose of this note is to observe that the causality assumptions related to such a choice in results used to establish global encoding of causality, due to Sakovich and Sormani [9] and to Burtscher and García-Heveling [2], can be weakened.

gr-qc↗
arXiv2023

Initial data rigidity results

We present several rigidity results for initial data sets motivated by the positive mass theorem. An important step in our proofs here is to establish conditions that ensure that a marginally outer trapped surface is "weakly outermost". A rigidity result for Riemannian manifolds with a lower bound on their scalar curvature is included as a special case.

gr-qc↗
arXiv2023

Some rigidity results for compact initial data sets

In this paper, we prove several rigidity results for compact initial data sets, in both the boundary and no boundary cases. In particular, under natural energy, boundary, and topological conditions, we obtain a global version of the main result in [15]. We also obtain some extensions of results in [12]. A number of examples are given in order to illustrate some of the results presented in this paper.

gr-qc↗
arXiv2022

The codimension 2 null cut locus with applications to spacetime topology

In this paper we review and extend some results in the literature pertaining to spacetime topology while naturally utilizing properties of the codimension 2 null cut locus. Our results fall into two classes, depending on whether or not one assumes the presence of horizons. Included among the spacetimes we consider are those that apply to the asymptotically (locally) AdS setting.

gr-qc↗
arXiv2022

Maximal hypersurfaces in asymptotically Anti-de Sitter spacetime

We prove uniqueness, existence, and regularity results for maximal hypersurfaces in spacetimes with a conformal completion at timelike infinity and asymptotically constant scalar curvature, as relevant for asymptotically AdS spacetimes. This work is dedicated to Yvonne Choquet-Bruhat on the occasion of her upcoming 99th birthday.

gr-qc↗
arXiv2022

The Topology of General Cosmological Models

Is the universe finite or infinite, and what shape does it have? These fundamental questions, of which relatively little is known, are typically studied within the context of the standard model of cosmology where the universe is assumed to be homogeneous and isotropic. Here we address the above questions in highly general cosmological models, with the only assumption being that the average flow of matter is irrotational. Using techniques from differential geometry, specifically extensions of the Bonnet-Myers theorem, we derive a condition which implies a finite universe and yields a bound for its diameter. Furthermore, under a weaker condition involving the interplay between curvature and diameter, together with the assumption that the universe is finite (i.e., has closed spatial slices), we provide a concise list of possible topologies. Namely, the spatial sections then would be either the ring topologies $S^1 \times S^2$, $S^1\tilde{\times}S^2$, $S^1\times\mathbb{RP}^2$, $\mathbb{RP}^3 \# \mathbb{RP}^3$, or covered by the sphere $S^3$ or torus $T^3$. In particular, under this condition the basic construction of connected sums would be ruled out (save for one), along with the plethora of topologies associated with negative curvature. These results are obtained from consequences of the geometrization of 3-manifolds, by applying a generalization of the almost splitting theorem together with a curvature formula of Ehlers and Ellis.

gr-qc↗