FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
equations and Mass-Momentum inequalities
May 11 - 15, 2015
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
Eric Poisson, University of Guelph
Dain, Universidad Nacional de Córdoba
Michael Holst,University of California, San Diego
210 , The Fields Institute
Lydia Bieri, University of Michigan
Spacetime Geometry and Radiation
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
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, 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
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.
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