Session abstracts

Session S19 - Geometric and Analytic Aspects of General Relativity



Monday, July 12, 16:00 ~ 17:00 UTC-3

Bakry-\'Emery Ricci curvature and general relativity

Eric Woolgar

University of Alberta, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The famous Penrose singularity theorem of 1965 depends on an energy condition which asserts the nonnegativity of null components of the Ricci tensor. Indeed, many important theorems in Riemannian geometry and relativity depend on a Ricci curvature bound. It was therefore surprising (to me) that often such theorems can be proved when the bounds only hold on the Ricci tensor modulo the hessian of some function or, more generally, the Lie derivative of the metric along some vector field. This is the Bakry-\'Emery Ricci tensor. The surprise is lessened only somewhat by the realization that Bakry-\'Emery Ricci curvature arises naturally in static spacetimes, near horizon geometries, Kaluza-Klein compactifications (warped products), and other natural applications. I will discuss a selection of these applications, including an application to the topology of closed universes (joint work with M Khuri and GJ Galloway) which yields topological constraints even when closure density is not quite achieved. Time permitting, I will also discuss synthetic Bakry-\'Emery Ricci tensor bounds in metric-measure spaces. Recent work by McCann and independently by Mondino and Suhr construct synthetic Ricci curvature bounds (energy conditions) in Lorentzian geodesic spaces. Cavalletti and Mondino have given a proof of a generalized Hawking cosmological singularity theorem for synthetic energy conditions. Burtscher, Ketterer, McCann, and I have given a Riemannian analogue.

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Monday, July 12, 17:00 ~ 18:00 UTC-3

Existence of static vacuum extensions

Lan-Hsuan Huang

University of Connecticut, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The study of static vacuum Riemannian metrics arises naturally in general relativity and differential geometry. A static vacuum metric produces a static spacetime by a warped product, and it is related to scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static vacuum metric with black hole boundary must belong to the Schwarzschild family. In contrast to the rigidity phenomenon, R. Bartnik conjectured that there are asymptotically flat, static vacuum metric realizing certain arbitrarily specified boundary data. I will discuss recent progress toward this conjecture.

Joint work with Zhongshan An (University of Connecticut).

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Monday, July 12, 18:00 ~ 19:00 UTC-3

On the construction of initial data sets

Armando Cabrera Pacheco

Universität Tübingen, Germany   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

From the point of view of the Cauchy problem for the Einstein equations, a fundamental problem is to understand and construct solutions to the so called constraint equations; those solutions are referred to as initial data sets. Although this is a very challenging problem, several methods exist to construct and study such solutions, under particular conditions. Remarkably, many of these methods are tailored to obtain time symmetric initial data sets. In this talk, we will give a short review of this problem and well known relevant results, we will then report some progress on a project focused, in part, to constructing initial data sets when time symmetry is not assumed.

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Monday, July 12, 19:00 ~ 20:00 UTC-3

Intrinsic flat convergence of points and manifolds and applications to stability of the positive mass theorem

Raquel Perales

IMATE UNAM, Oaxaca , Mexico   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk we will revisit the intrinsic flat stability result for the case of graphical hypersurfaces of Euclidean space shown by Huang-Lee-Sormani and will explain how to prove it using new results by Allen-Perales and Huang-Lee-Perales. These new results consist of a convergence theorem with respect to the intrinsic flat distance and the compatibility of the convergence of sequences of points whenever one has intrinsic flat and Gromov-Hausdorff convergence of a sequence of manifolds.

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Thursday, July 15, 16:00 ~ 17:00 UTC-3

Fillins with nonnegative scalar curvature

Christos Mantoulidis

Rice University and Brown University, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We discuss the problem of finding a compact Riemannian manifold with mean-convex boundary and nonnegative scalar curvature, given a prescribed metric on the boundary. This problem relates to quasi-local mass, and despite some important recent progress, many basic open questions remain.

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Thursday, July 15, 17:00 ~ 18:00 UTC-3

Causality on coverings and the Gannon-Lee theorem

Ivan Costa e Silva

Federal University of Santa Catarina, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

A number of techniques in Lorentzian geometry, such as those used in the proofs of singularity theorems, depend on certain smooth coverings retaining interesting global geometric properties, including causal ones. We briefly review this prolem and give explicit examples showing that, unlike some of the more commonly adopted rungs of the causal ladder such as strong causality or global hyperbolicity, less-utilized conditions such as causal continuity or causal simplicity {\em do not} in general pass to coverings, as already speculated by E. Minguzzi. As a consequence, any result which relies on these causality conditions transferring to coverings must be revised accordingly. In particular, we show concrete applications of these ideas to the Gannon-Lee singularity theorem.

Joint work with Ettore Minguzzi (University of Florence, Italy).

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Thursday, July 15, 18:00 ~ 19:00 UTC-3

Brill-Lindquist-Riemann sums and their limits

Iva Stavrov

Lewis & Clark College, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk we introduce discretized, point-source, relativistic initial data, called Brill-Lindquist-Riemann sums, and examine their convergence towards a charged dust continuum. We are motivated by the interpretation of Brill-Lindquist-Riemann sums as collections of relatively isolated astrophysical bodies such as stars and galaxies in the universe, and the interpretation of the dust continuum as the universe itself. We present the existence and the uniqueness results for horizons/minimal surfaces of Brill-Lindquist metrics in the vicinity of the point-sources ("stars"). We also study the geometries of the regions exterior to said minimal surfaces, and discuss their Gromov-Hausdorff and intrinsic flat limits.

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Thursday, July 15, 19:00 ~ 20:00 UTC-3

Initial data rigidity results

Abraão Mendes

Universidade Federal de Alagoas, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In this lecture we aim to present some rigidity results for initial data sets that are motivated by the spacetime positive mass theorem. A key step is to show that certain marginally outer trapped surfaces (MOTS) are weakly outermost. As a special case, our results include a rigidity result for Riemannian manifolds with a lower bound on their scalar curvature.

Joint work with Michael Eichmair (University of Vienna, Austria) and Gregory J. Galloway (University of Miami, USA).

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Wednesday, July 21, 16:00 ~ 17:00 UTC-3

Lower Bounds for the Total Mass in 3-Dimensions

Marcus Khuri

Stony Brook University, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We provide lower bounds for the total mass of 3-dimensional initial data sets that is based on (spacetime) harmonic functions. The technique works for both the asymptotically flat and asymptotically hyperboloidal settings. These bounds are valid without the assumption of nonnegative scalar curvature or the dominant energy condition. However, if the energy condition is assumed then the result yields a new proof of the positive mass theorem.

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Wednesday, July 21, 17:00 ~ 18:00 UTC-3

Integrable structures in 4-manifolds

Bernardo Araneda

Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Germany   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Inspired by recent progress on the geometrical understanding of T-duality in string theory, we apply techniques from generalized and (para-)complex geometry to the analysis of different kinds of integrable structures in 4-manifolds equipped with a metric of Lorentzian, Riemannian, or split signature, with emphasis on applications to general relativity and twistor theory. In particular, we classify all (possibly complex-valued) almost para-Hermitian structures, we analyse their integrability, and we describe related constructions such as Lie and Courant algebroids, twisted de Rham complexes, deformations of (para-)complex structures, and two- and three-dimensional twistor spaces.

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Wednesday, July 21, 18:00 ~ 19:00 UTC-3

The conformal method applied to fluids (done right)

David Maxwell

University of Alaska Fairbanks, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The conformal method has long been a mainstay for the construction of initial data in general relativity. Although the method has been used productively to generate non-vacuum initial data for a number of matter models based on carefully choosing a scaling for `seed data' associated with the matter fields, the scaling begged the question of what exactly was being specified about the finally constructed initial data. In recent joint work with Isenberg, we exhibited an underlying principle that determines for a given matter model how to scale it and what is being specified. In effect, a matter field and its conjugate momentum are chosen explicitly and do not scale.

In this talk, we show how to apply this principle to perfect fluids. This is an interesting case, in part because the equations obtained differ from those found in the past by ad-hoc methods. The presentation is based on a careful analysis of the Lagrangian for fluids, and the talk will include an elementary exposition of this topic.

Joint work with Jim Isenberg (University of Oregon).

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Wednesday, July 21, 19:00 ~ 20:00 UTC-3

Initial Boundary Value Problem For Vacuum Einstein Equations

Zhongshan An

University of Connecticut, United States of America

In general relativity, spacetime metrics satisfy the Einstein equations, which are wave equations in the harmonic gauge. The Cauchy problem for the vacuum Einstein equations has been well-understood since the work of Choquet-Bruhat. For an initial data set satisfying the vacuum constraint equations, there exists a solution to the vacuum Einstein equations and it is geometrically unique in the domain of dependence of the initial surface. On contrast, the initial boundary value problem (IBVP) has been much less understood. To solve for an vacuum metric in a region with time-like boundary, one needs to impose boundary conditions to guarantee geometric uniqueness of the solution. However, due to gauge issues occurring on the boundary, there has not been a satisfying choice of boundary conditions. In this talk I will discuss obstacles in establishing a well-defined IBVP for vacuum Einstein equations and the geometric uniqueness problem. Then I will talk about some new results in a joint work with Michael Anderson.

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Strong cosmic censorship theorem in Bakry-Emery spacetimes

Makoto Narita

National Institute of Technology, Okinawa College, Japan   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

A class of naked strong curvature singularities is ruled out in Bakry-Emery spacetimes by using techniques of differential topology in Lorentzian manifolds. These spacetimes adimit a Bakry-Emery-Ricci tensor which is a generalization of the Ricci tensor. This result supports to validity of Penrose's strong cosmic censorship conjecture in scalar-tensor gravitational theories, which include dilaton gravity and Brans-Dicke theory.

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