# SCIENTIFIC PROGRAMS AND ACTIVITIES

January 27, 2022

THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

## Constraint equations and Mass-Momentum inequalities May 11 - 15, 2015

 Organizing Committee Focus Week Organizers Location Spyros Alexakis, University of Toronto Mihalis Dafermos, Princeton University Luis Lehner, Perimeter Institute for Theoretical Physics and University of Guelph Harald Pfeiffer, Canadian Institute for Theoretical Astrophysics (CITA) Eric Poisson, University of Guelph Sergio Dain, Universidad Nacional de Córdoba Michael Holst,University of California, San Diego Mu-Tao Wang, Columbia University Room 210 , The Fields Institute

Abstracts

Lydia Bieri, University of Michigan

In General Relativity, a major branch of research is devoted to the study of the geometric properties of solutions to the Einstein equations. Gravitational waves, which are predicted by the theory of General Relativity and expected to be detected in the near future, are fluctuations of the spacetime curvature. These waves leave a footprint in the spacetime regions they travelled through. We investigate the geometric-analytic properties of various spacetimes with gravitational radiation.

Carla Cederbaum, Universität Tübingen

Uniqueness of static photon spheres

We show that the Schwarzschild spacetimes of positive mass are the only static vacuum asymptotically flat general relativistic spacetimes that possess a suitably geometrically defined photon sphere. We will present two proofs, both extending classical static black hole uniqueness results. Part of this work is joint with Gregory J. Galloway. As a corollary, we obtain a new result concerning the static n-body problem.

Sergio Dain, Universidad Nacional de Córdoba

Geometric inequalities for black holes and bodies

A geometric inequality in General Relativity relates quantities that have both a physical interpretation and a geometrical definition. It is well known that the parameters that characterize the Kerr-Newman black hole (angular momentum, charge, mass and horizon area) satisfy several important geometric inequalities. Remarkably enough, some of these inequalities also hold for dynamical black holes. This kind of inequalities, which are valid in the dynamical and strong field regime, play an important role in the characterization of the gravitational collapse. They are closed related with the cosmic censorship conjecture. Also, variants of these inequalities are valid for ordinary bodies.

In these two talks I will give an overview of the subject. The first talk will be focused on black holes and the second on bodies.

Michael Holst, University of California, San Diego

Overview of the analysis frameworks for non-CMC solutions to the conformal method equations (Part I)

In this overview lecture, we begin with a brief overview of the 1973-1974 conformal method. We then summarize the CMC (constant mean curvature) and near-CMC results that had been established during the period 1973 through 2007. We then give an overview of the new framework that was developed in 2008 for removing the near-CMC condition, and outline the generalizations made to the framework from 2009 to 2013 (vacuum, rough metrics, manifolds with boundary, AE manifolds, and the limit equation). We then describe two interesting developments that have substantially changed the direction of the field: the emergence of degeneracies in the so-called far-from-CMC cases (beginning in 2010), and the observation that some non-CMC results can be obtained with implicit function arguments around zero mean curvature, without resorting to near-CMC conditions.

Overview of the analysis frameworks for non-CMC solutions to the conformal method equations (Part II)

Picking up from the first overview lecture, we look a little more closely at some of the non-CMC results for the conformal method that began to appear in 2008, and examine results for closed manifolds from
2008-2009, compact manifolds with boundary from 2013-2014, and asymptotically Euclidean manifolds from 2014-2015. We give a summary of the results for rough metrics in each of these cases through 2015, and describe some results that examine non-uniqueness in the non-CMC case through the use of analytic bifurcation theory.

Greg Galloway, University of Miami

Rigidity of marginally outer trapped 2-spheres

In a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inquality. Namely, as will be discussed, its area must be bounded above by $4\pi/c$, where $c > 0$ is a lower bound on a natural energy-momentum term. We then consider the rigidity that results for stable, or weakly outermost, marginally outer trapped 2-spheres that achieve this upper bound on the area. The results obtained are spacetime generalizations of the rigidity results of Bray, Brendle and Neves concerning area minimizing 2-spheres in Riemannian 3-manifolds with scalar curvature having positive lower bound. Connections to Vaidya spacetime and Nariai spacetime are discussed. The talk is based primarily on joint work with Abraao Mendes.

Pei-Ken Hung, Columbia University

Gibbons-Penrose inequality

Gibbons-Penrose inequality, like Penrose inequality, is a conjecture proposed by Penrose as a test of weak consmic censorship. I will discuss the heuristic argument of Penrose and some work in this direction.

Marcus Khuri, Stony Brook University

A Mass-Angular Momentum-Charge Inequality for Multiple Black Holes, Size-Angular Momentum-Charge Inequalities, and Existence of Black Holes

In the first part of the talk we present a proof of the mass-angular momentum-charge inequality for multiple black holes (joint with Gilbert Weinstein). In the second part, new inequalities relating the size and angular momentum as well as size and charge of bodies is presented. Lastly, black hole existence results due to concentration of angular momentum and charge will be discussed.

Dan Lee, Queens College, CUNY

Stability of the positive mass theorem

The rigidity of the positive mass theorem is the fact that Euclidean space is the only complete asymptotically flat manifold with nonnegative scalar curvature and zero mass. I will survey some work on the stability of this statement.

Walter Simon, Universität Wien

Initial data for rotating cosmologies Initial data for rotating cosmologies

Using the conformal method I recall the construction of "t-phi-symmetric" initial data on compact manifolds in vacuum with positive cosmological constant. I discuss a recent, key result of Premoselli which yields (non-)existence, (non-)uniqueness and (in-)stability of solutions of the corresponding Lichnerowicz equation, depending on its coefficients. We (joint work with Piotr Bizon and Stefan Pletka) apply this result to examples which demonstrate the role of the angular momentum for the existence problem, as well as the relation between stability and symmetry of the solutions.

Gantumur Tsogtgerel, McGill University

On the Lichnerowicz equation and the prescribed scalar-mean curvature problem in the compact-with-boundary setting

In the first half of this talk, I will discuss some details of our recent work on the existence theory of the Lichnerowicz equation on compact manifolds with boundary. This is a joint work with Michael Holst. Then the second half of the talk will be concerned with a related problem of prescribing scalar curvature and boundary mean curvature of a compact manifold with boundary. This is an ongoing work and builds on the results of Rauzy and of Dilts-Maxwell.

Mu-Tao Wang, Columbia University

Conserved quantities in general relativity, Part I and II

I shall discuss conserved quantities such as energy, linear momentum, angular momentum, and center of mass. Specifically, the topics will include:
1. Reviews of classical notions for isolated systems.
2. Recent progresses of defining these notions at the quasi-local level.