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Eric Woolgar

Publications and source records attributed to Eric Woolgar.

At least 19 recordsLinked to original sources
arXiv2026

Conformal boundary rigidity from null geodesic travel times

The gravitational field of a distant, isolated system is manifested by the conformally invariant Weyl tensor. Thus the conformal structure far from the system encodes the system's gravitational mass. It also encodes the causal structure, thereby linking it to the mass. For asymptotically anti-de Sitter (AdS) spacetimes, this link led to a novel positive mass theorem of Page, Surya, and the second author \cite{PSW} which did not rely on any traditional energy condition. Here we ask whether that theorem has a rigidity case. Specifically, we consider all null geodesics in an asymptotically AdS spacetime that depart from the Penrose conformal infinity, travel through spacetime, and return to conformal infinity. If all such geodesics from a given point refocus at an antipodal point at infinity, is the spacetime conformal to anti-de Sitter space? It is easy to answer the question if the asymptotically AdS spacetime either (i) obeys the null energy condition in 3 or 4 spacetime dimensions, or (ii) is static (in any dimension), and we give simple proofs in those cases. We then answer the question in the case of globally stationary, asymptotically AdS spacetimes, by applying the theory of magnetic geodesics on the Riemannian manifold-with-boundary obtained by quotienting by the stationary Killing vector field. The question has an analogue for asymptotically flat spacetimes, which we also discuss.

math.DG↗
arXiv2025

The Penrose singularity theorem, MOTS stability, and horizon topology in weighted spacetimes

We consider versions of the Penrose singularity theorem and the Hawking horizon topology theorem in weighted spacetimes that contain weighted versions of trapped surfaces, for arbitrary spacetime dimension and synthetic dimension. We find that suitable generalizations of the unweighted theorems hold under a weighted null energy condition. Our results also provide further evidence in favour of a weighted scalar curvature that differs from the trace of the weighted Ricci curvature. When the synthetic dimension is a positive integer, these weighted curvatures have a natural interpretation in terms of warped product metrics.

gr-qc↗
arXiv2023

Deformations of the Kerr-(A)dS Near Horizon Geometry

We investigate deformations of the Kerr-(A)dS near horizon geometry and derive partial infinitesimal rigidity results for it. The proof comprises two parts. First, we follow the analysis of Jezierski and Kami\'nski [Gen Rel Grav 45 (2013) 987--1004] to eliminate all but a finite number of Fourier modes of linear perturbations. In the second part, we give an argument using analyticity to prove that there are no odd Fourier modes.

gr-qc↗
arXiv2023

Rigidity of quasi-Einstein metrics: The incompressible case

As part of a programme to classify quasi-Einstein metrics $(M,g,X)$ on closed manifolds and near-horizon geometries of extreme black holes, we study such spaces when the vector field $X$ is divergence-free but not identically zero. This condition is satisfied by left-invariant quasi-Einstein metrics on compact homogeneous spaces (including the near-horizon geometry of an extreme Myers-Perry black hole with equal angular momenta in two distinct planes), and on certain bundles over K\"ahler-Einstein manifolds. We find that these spaces exhibit a mild form of rigidity: they always admit a one-parameter group of isometries generated by $X$. Further geometrical and topological restrictions are also obtained.

math.DG↗
arXiv2022

Static near horizon geometries and rigidity of quasi-Einstein manifolds

Static vacuum near horizon geometries are solutions $(M,g,X)$ of a certain quasi-Einstein equation on a closed manifold $M$, where $g$ is a Riemannian metric and $X$ is a closed 1-form. It is known that when the cosmological constant vanishes, there is rigidity: $X$ vanishes and consequently $g$ is Ricci flat. We study this form of rigidity for all signs of the cosmological constant. It has been asserted that this rigidity also holds when the cosmological constant is negative, but we exhibit a counter-example. We show that for negative cosmological constant if $X$ does not vanish identically, it must be incompressible, have constant norm, and be nontrivial in cohomology, and $(M,g)$ must have constant scalar curvature and zero Euler characteristic. If the cosmological constant is positive, $X$ must be exact (and vanishing if $\dim M=2$). Our results apply more generally to a broad class of quasi-Einstein equations on closed manifolds. We extend some known results for quasi-Einstein metrics with exact 1-form $X$ to the closed $X$ case. We consider near horizon geometries for which the vacuum condition is relaxed somewhat to allow for the presence of a limited class of matter fields. An appendix contains a generalization of a result of Lucietti on the Yamabe type of quasi-Einstein compact metrics (with arbitrary $X$).

math.DG↗
arXiv2021

Bakry-\'Emery Ricci Curvature Bounds on Manifolds with Boundary

We prove a Bakry-\'Emery generalization of a theorem of Petersen and Wilhelm, itself a generalization of a theorem of Frankel, that closed minimal hypersurfaces in a complete manifold with a suitable curvature bound must intersect. We then prove splitting theorems of Croke-Kleiner type for manifolds bounded by hypersurfaces obeying Bakry-\'Emery curvature bounds. Motivated in part by the near-horizon geometry programme of general relativity, we do not assume that the Bakry-\'Emery vector field is of gradient type.

math.DG↗
arXiv2021

Self-similar curve shortening flow in hyperbolic 2-space

We find and classify self-similar solutions of the curve shortening flow in standard hyperbolic 2-space. Together with earlier work of Halld\'orsson on curve shortening flow in the plane and Santos dos Reis and Tenenblat in the 2-sphere, this completes the classification of self-similar curve shortening flows in the constant curvature model spaces in 2-dimensions.

math.DG↗
arXiv2020

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↗
arXiv2020

A Bakry-\'Emery Almost Splitting Result With Applications to the Topology of Black Holes

The almost splitting theorem of Cheeger-Colding is established in the setting of almost nonnegative generalized $m$-Bakry-\'{E}mery Ricci curvature, in which $m$ is positive and the associated vector field is not necessarily required to be the gradient of a function. In this context it is shown that with a diameter upper bound and volume lower bound the fundamental group of such manifolds is almost abelian. Furthermore, extensions of well-known results concerning Ricci curvature lower bounds are given for generalized $m$-Bakry-\'{E}mery Ricci curvature. These include: the first Betti number bound of Gromov and Gallot, Anderson's finiteness of fundamental group isomorphism types, volume comparison, the Abresch-Gromoll inequality, and a Cheng-Yau gradient estimate. Finally, this analysis is applied to stationary vacuum black holes in higher dimensions to find that low temperature horizons must have limited topology, similar to the restrictions exhibited by (extreme) horizons of zero temperature.

math.DG↗
arXiv2020

Inscribed Radius Bounds for Lower Ricci Bounded Metric Measure Spaces with Mean Convex Boundary

Consider an essentially nonbranching metric measure space with the measure contraction property of Ohta and Sturm, or with a Ricci curvature lower bound in the sense of Lott, Sturm and Villani. We prove a sharp upper bound on the inscribed radius of any subset whose boundary has a suitably signed lower bound on its generalized mean curvature. This provides a nonsmooth analog to a result of Kasue (1983) and Li (2014). We prove a stability statement concerning such bounds and - in the Riemannian curvature-dimension (RCD) setting - characterize the cases of equality.

math.DG↗
arXiv2018

Formal power series for asymptotically hyperbolic Bach-flat metrics

It has been observed by Maldacena that one can extract asymptotically anti-de Sitter Einstein $4$-metrics from Bach-flat spacetimes by imposing simple principles and data choices. We cast this problem in a conformally compact Riemannian setting. Following an approach pioneered by Fefferman and Graham for the Einstein equation, we find formal power series for conformally compactifiable, asymptotically hyperbolic Bach-flat 4-metrics expanded about conformal infinity. We also consider Bach-flat metrics in the special case of constant scalar curvature and in the special case of constant $Q$-curvature. This allows us to determine the free data at conformal infinity, and to select those choices that lead to Einstein metrics. Interestingly, the mass is part of that free data, in contrast to the pure Einstein case. We then choose a convenient generalization of the Bach tensor to (bulk) dimensions $n>4$ and consider the higher dimensional problem. We find that the free data for the expansions split into low-order and high-order pairs. The former pair consists of the metric on the conformal boundary and its first radial derivative, while the latter pair consists of the radial derivatives of order $n-2$ and $n-1$. Higher dimensional generalizations of the Bach tensor lack some of the geometrical meaning of the 4-dimensional case. This is reflected in the relative complexity of the higher dimensional problem, but we are able to obtain a relatively complete result if conformal infinity is not scalar flat.

math.DG↗
arXiv2018

New Restrictions on the Topology of Extreme Black Holes

We provide bounds on the first Betti number and structure results for the fundamental group of horizon cross-sections for extreme stationary vacuum black holes in arbitrary dimension, without additional symmetry hypotheses. This is achieved by exploiting a correspondence between the associated near-horizon geometries and the mathematical notion of $m$-quasi Einstein metrics, in addition to generalizations of the classical splitting theorem from Riemannian geometry. Consequences are analyzed and refined classifications are given for the possible topologies of these black holes.

hep-th↗
arXiv2017

Nonexistence of Degenerate Horizons in Static Vacua and Black Hole Uniqueness

We show that in any spacetime dimension $D\ge 4$, degenerate components of the event horizon do not exist in static vacuum configurations with positive cosmological constant. We also show that without a cosmological constant asymptotically flat solutions cannot possess a degenerate horizon component. Several independent proofs are presented. One proof follows easily from differential geometry in the near-horizon limit, while others use Bakry-\'Emery-Ricci bounds for static Einstein manifolds.

gr-qc↗
arXiv2017

Nonexistence of Extremal de Sitter Black Rings

We show that near-horizon geometries in the presence of a positive cosmological constant cannot exist with ring topology. In particular, de Sitter black rings with vanishing surface gravity do not exist. Our result relies on a known mathematical theorem which is a straightforward consequence of a type of energy condition for a modified Ricci tensor, similar to the curvature-dimension conditions for the $m$-Bakry-\'Emery-Ricci tensor.

hep-th↗
arXiv2017

Curvature-dimension bounds for Lorentzian splitting theorems

We analyze Lorentzian spacetimes subject to curvature-dimension bounds using the Bakry-\'Emery-Ricci tensor. We extend the Hawking-Penrose type singularity theorem and the Lorentzian timelike splitting theorem to synthetic dimensions $N\le 1$, including all negative synthetic dimensions. The rigidity of the timelike splitting reduces to a warped product splitting when $N=1$. We also extend the null splitting theorem of Lorentzian geometry, showing that it holds under a null curvature-dimension bound on the Bakry-\'Emery-Ricci tensor for all $N\in (-\infty, 2]\cup (n,\infty)$ and for the $N=\infty$ case as well, with reduced rigidity if $N=2$. In consequence, the basic singularity and splitting theorems of Lorentzian Bakry-\'Emery theory now cover all synthetic dimensions for which such theorems are possible. The splitting theorems are found always to exhibit reduced rigidity at the critical synthetic dimension.

math.DG↗
arXiv2016

Ricci Flow and Volume Renormalizability

With respect to any special boundary defining function, a conformally compact asymptotically hyperbolic metric has an asymptotic expansion near its conformal infinity. If this expansion is even to a certain order and satisfies one extra condition, then it is possible to define its renormalized volume and show that it is independent of choices that preserve this evenness structure. We prove that such expansions are preserved under normalized Ricci flow. We also study the variation of curvature functionals in this setting, and as one application, obtain the variation formula $$ \frac{{\rm d}}{{\rm d}t} {\rm RenV}\big(M^n, g(t)\big) = -\mathop{\vphantom{T}}^R \! \! \! \int_{M^n} (S(g(t))+n(n-1)) {\rm d}V_{g(t)},$$ where $S(g(t))$ is the scalar curvature for the evolving metric $g(t)$, and $\mathop{\vphantom{T}}^R \! \! \! \int (\cdot) {\rm d}V_g$ is Riesz renormalization. This extends our earlier work to a broader class of metrics.

math.DG↗
arXiv2016

The rigid Horowitz-Myers conjecture

The "new positive energy conjecture" Horowitz and Myers (1999) probes a possible nonsupersymmetric AdS/CFT correspondence. We consider a version formulated for complete, asymptotically Poincar\'e-Einstein Riemannian metrics $(M,g)$ with bounded scalar curvature $R\ge -n(n-1)$. This version then asserts that any such $(M,g)$ must have mass not less than the mass $m_0$ of a metric $g_0$ induced on a time-symmetric slice of a certain AdS soliton spacetime. The conjecture remains unproved, having so far resisted standard techniques. Little is known other than that the conjecture is true for metrics which are sufficiently small perturbations of $g_0$. We pose another test for the conjecture. We assume its validity and attempt to prove as a corollary the corresponding scalar curvature rigidity statement, that $g_0$ is the unique asymptotically Poincar\'e-Einstein metric with mass $m=m_0$ obeying $R\ge -n(n-1)$. Were a second such metric $g_1$ not isometric to $g_0$ to exist, it then may well admit perturbations of lower mass, contradicting the assumed validity of the conjecture. We find that the minimum mass metric must be static Einstein, so the problem is reduced to that of static uniqueness. When $n=3$ the manifold is isometric to a time-symmetric slice of an AdS soliton spacetime, unless it has a non-compact horizon. En route we study the mass aspect, obtaining and generalizing known results. The mass aspect is (i) related to the holographic energy density, (ii) a weighted invariant under boundary conformal transformations when the bulk dimension is odd, and (iii) zero for negative Einstein manifolds with Einstein conformal boundary.

math.DG↗