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Speaker
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Title/Abstract
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Date and Time
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Week 3
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J.Kijowski
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Black hole thermodynamics and the Penrose
inequality
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Tuesday June 17, 16.00
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Abstract:
Geometric objects describing internal and external geometry of a
null-like hypersurface (``wave front'') are discussed. Hamiltonian
formula generating time evolution of the gravitational field
outside of such a wave front is derived. It gives us variation of
the total field energy (the ADM mass) with respect to both the
volume degrees of freedom of the field and its boundary data on
the wave front. The latter are represented by the internal and
external geometry of the front. A special case of a non-expanding
horizon is discussed in detail. Field configurations for which
variation of the energy with respect to volume degrees of freedom
vanish are black (or white) holes. It is shown that in this case
the Hamiltonian formula reduces to the so called ``first law of
thermodynamics'' of black holes. It is conjectured that the energy
is a convex functional. This would imply the following inequality:
energy (mass) of a black hole is minimal among all the (dynamical)
field configurations having the same boundary data on the horizon.
A similar relation between dynamics and ``thermodynamics of
stationary solutions'' is discussed in other field theories (wave
equation, Maxwell electrodynamics) and the analog of the above
Penrose inequality is proved.
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N.O'Murchadha
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On the Schoen-Yau positivity proof
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Wednesday June 18, 13.30
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Abstract:
Given an asymptotically flat Riemannian three-manifold with negative mass
and
vanishing scalar curvature, I show, using an elementary matching technique,
that
one can construct a manifold which has nonnegative scalar curvature and is
flat in
the exterior. One can now construct a cube, by putting the walls
in the flat region, that encloses the nontrivial part of the manifold.
By an appropriate identification, this cube can be converted into a
three-torus with
positive scalar curvature. Such an object cannot exist (Gromov - Lawson
etc.). Therefore
the initial assumption of negative mass cannot be correct. This is a
(significant)
simplification of the first of the Schoen and Yau positivity proofs.
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E.Malec
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The Penrose inequality and the quasi-local
mass
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Friday June 20, 14.00
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Abstract: Some existing schemes for the proving of the Penrose
inequality are reviewed. The basic ingredient of these
is a special foliation satisfying the Geroch condition
and some other requirements. The Hawking quasilocal mass
appears to be a sensible quantity in those foliations.
A volume representation of the Hawking mass is given.
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Week 4
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| M.Herzlich
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A Penrose-type inequality
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Monday June 23, 16.00
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Abstract:
The talk will mainly survey the spinorial approaches to positive mass
theorems or Penrose inequalities, focussing on the author's own work
on the subject, dating back to 1997. Although this method didn't lead to
proofs of the expected inequality, one might hope that it will yield new results in the future.
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| K.Roszkowski
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Herzlich's inequality in Reissner-Nordstroem metrics
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Tuesday June 24, 13.30-13.50
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Abstract:
We show that the constant appearing in Herzlich's inequality tends to zero for charged black holes approaching
the extremal Reissner-Nordstroem solutions.
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| G.Bergqvist
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Monotone quantities for the Penrose inequality
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Tuesday June 24, 14.00
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Abstract:
We discuss possibilities of using the Nester-Witten
integral when studying the Penrose inequality.
Advantages are its monotonicity properties while a
problem is to relate it to the surface area. We
compare this approach with that of using the
Hawking mass.
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| R.Schoen
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The Vacuum Constraint
Equations and Generalized Penrose Inequalities
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Wednesday June 25, 11.00
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Abstract:In this talk we will discuss a particularly special form for the
asymptotic behavior of solutions of the asymptotically flat vacuum
constraint equations. We will explain why this form is achievable in the
sense that solutions with these asymptotics form a dense set (in a
suitably chosen topology) in the set of all solutions. We use this form to
give a new, more geometric, proof of the timelike property of the ADM
energy-momentum vector, and we discuss the generalized positive mass and
Penrose conjectures involving angular momentum.
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| H.Bray
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On the Penrose inequality I
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Thursday June 26, 14.00
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Abstract:In this series of talks we will discuss both the Huisken-Ilmanen and
Bray proofs of the Riemannian Penrose Inequality, the challenging
technical issues which occur in higher dimensions (n > 7) for both the
Riemannian Penrose Inequality and the Positive Mass Theorem, and
approaches to proving the full Penrose Conjecture. The talks will attempt
to both summarize previous results as well as try to list the interesting
approaches and conjectures which are still open.
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| H.Bray
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On the Penrose inequality II
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Friday June 27, 15.00
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Week 5
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| H.Bray
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On the Penrose inequality III
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Monday June 30, 10.30
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| H.Bray
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On the Penrose inequality IV
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Tuesday July 1, 10.30
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| J.Lohkamp
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Positive Scalar Curvature I
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Tuesday July 1, 16.00-17.30
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Abstract:
There are two by now nearly classical approaches to (obstruction theory
for) positive scalar curvature. Spin Geometry (initiated by Lichnerowicz,
Gromov, Lawson, Witten) and the inductive argument via minimal hypersurfaces
introduced by Schoen and Yau. Whereas the first approach is restricted to
spin manifolds, the second one covers arbitrary topologies. But there is
one serious drawback: Due to the appearance of singularities the method could
be applied only in dimension less or equal 7. We want to report about recent
progress how to overcome this dimensional obstacle by some local stabilisation
process and indicate how to extend this approach to more general situations
including higher dimensions. |
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| J.Lohkamp
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Positive Scalar Curvature II
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Wednesday July 2, 11.00-12.30
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| M.Aarons
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Mean curvature flow of spacelike hypersurfaces
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Thursday July 3, 16.00
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Abstract: Ecker and Huisken have shown that the mean curvature flow
(MCF) can be used to construct prescribed mean curvature hypersurfaces in
cosmological spacetimes. We aim to generalise this result to
asymptotically flat spacetimes (which is more difficult due to
non-compactness.) In this talk, I will present such a result for
Minkowski space, and point-out how the results may generalise to a curved
background (which is the subject of current work). In particular, I will
present sketch-proofs showing that MCF: (1) exists for all
time; (2) converges globally to a prescribed mean curvature hypersurface
(provided the boundary conditions at infinity are appropriate); and
(3) that all hypersurfaces become convex under the flow.
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| J.Jezierski
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Penrose-type inequalities and p-harmonic gauge
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Friday July 4, 16.00
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Abstract: Assuming the existence of a solution to the p-harmonic equation $\nabla_i
(|\nabla\rho|^p \nabla^i\rho)=0$ on a spacelike three-manifold
Sigma with boundary K, we construct an integral identity.
The boundary term (at infinity) of this integrand is ADM mass at infinity.
The global integrand is manifestly positive (for some choices of Sigma).
The boundary term on K gives some number (when integrated over K)
related to the area of the horizon but in general differs from it.
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Week 6
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| S.Dain
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Trapped surfaces as boundaries for the constraint equations |
Monday July 7, 16.00
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Abstract:
In this talk I will discuss trapped surfaces as inner boundary for the
Einstein vacuum constraint equations. The trapped surface condition can
be written as a non linear boundary condition for these equations.
Under appropriate assumptions, I will prove existence and uniqueness
of solutions in the exterior region for this boundary value problem.
I will also discuss the relevance of this result for the black hole
collision problem.
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| G.Galloway
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Remarks on the positivity of mass for asymptotically AdS
spacetimes.
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Tuesday July 8, 16.00
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Abstract: We discuss an approach to the proof of positivity of mass,
without spin assumption, for asymptotically AdS spacetimes,
based on the general methodology of Schoen and Yau. Our
approach makes use of the "brane action" introduced by
Witten and Yau in their work on the AdS/CFT correspondence, and
takes hints from the work of Lohkamp. This is joint work with
Lars Andersson and Mingliang Cai.
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| T.Ilmanen
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Smooth computations for the Inverse Mean Curvature Flow (IMCF)
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Wednesday July 9, 16.00
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Abstract:
We present the estimates for classical quantities for IMCF, involving the
mean curvature, the second fundamental form, the Harnack inequality, and
the star-shaped quantity. |
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| T.Ilmanen
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Weak Formulation of the IMCF
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Thursday July 10, 16.00
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Abstract: We give a tutorial on the variational formulation of IMCF including weak existence theorem,
regularity theory, and passage to limits.
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| T.Ilmanen
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Topology, Infinity, etc.
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Friday July 11, 11.00
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Abstract: We discuss the topology of the evolving surfaces and the
approach to infinity in asymptotically flat and other settings.
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| T.Ilmanen
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Discussion
session, with programmed short contributions by T.Ilmanen,
H.Bray, W.Simon and several unprogrammed contributions by the members of
the audience. |
Friday July 11, 14.00
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Week 7
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| S.Hayward
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Penrose and other inequalities for black holes in or near spherical
symmetry
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Monday July 14, 16.00
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Abstract:
(i) Spherical symmetry allows a general proof of the Penrose inequality
for
dynamically evolving black holes. This holds not only for the ADM mass
at
spatial infinity, but for the (generally smaller) Bondi mass at null
infinity,
as required by cosmic censorship. In fact it holds for the local
gravitational
mass anywhere in a non-trapped region achronally outward from a marginal
surface, locally defining a black hole. There are other physically
relevant
inequalities involving surface gravity. The basic assumption is Einstein
gravity with the dominant energy condition. The proofs, using a
characteristic
or dual-null method, are mathematically simple and physically
understandable.
(ii) Generalizing beyond spherical symmetry, a result of Malec et al. is
confirmed: the Hawking mass on uniformly expanding flows is monotonic
under a
certain condition, and so satisfies the Penrose inequality achronally
outwards
from a marginal surface. (iii) To describe space-times close to
spherical
symmetry, a quasi-spherical approximation scheme has been developed. A
gravitational mass arises as the charge associated with a conserved
energy-momentum current, including the effective energy-momentum of
gravitational radiation. The mass satisfies local versions of the
Penrose
inequality and the other inequalities involving surface gravity.
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| M.Mars
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On the monotonicity of the Hawking mass
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Tuesday July 15, 16.00
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Abstract: We discuss how the Hawking quasi-local mass changes along
arbitrary flows in spacetime. We find necessary conditions
on the spacetime and on the flow to ensure monotonicity of the
Hawking mass. Local existence of such monotonic flows is proven
in the case of null velocity vector and some problems related
with local existence in the spacelike case are discussed.
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| R.Wald
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A General Analysis of Prescriptions for Defining Dynamics
in Non-Globally-Hyperbolic, Static Spacetimes
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Wednesday July 16, 16.00
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Abstract: Over twenty years ago, I proposed a prescription for defining
the dynamics of a scalar field in static, non-globally-hyperbolic
spacetimes. This prescription involved viewing the spatial derivative part
of the wave operator as an operator on a certain Hilbert space and
defining dynamics by taking the Friedrichs extension (or other positive
extension) of the operator. However, this proposal left open the
possibility that there could exist other satisfactory prescriptions for
defining dynamics that are of a totally different nature. In this talk, I
will describe recent work with A. Ishibashi that shows that, in fact, the
previously given prescription is the only one that agrees locally with the
dynamics defined by the wave equation, admits a globally conserved energy,
and satisfies certain other requirements.
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| R.Bartnik
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Bondi mass under weak decay conditions
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Thursday July 17, 16.00
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Abstract:
Using the null quasi-spherical gauge, we show that the Bondi mass is
well-defined and satisfies the Trautman-Bondi massloss formula, assuming
only a rather weak decay condition on just two of the metric coefficients,
and decay of most of the stress-energy tensor. The definition is local to
a single outgoing null hypersurface, and does not require any normalising
condition on a null foliation.
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| G.Weinstein
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On a Penrose Inequality with Charge
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Friday July 18, 16.00
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Abstract:
(Joint work with Sumio Yamada) We construct a time-symmetric asymptotically flat initial data set
to the Einstein-Maxwell Equations which satisfies
where m is the total mass,  is
the area radius of the
outermost horizon and Q is the total charge. This yields a
counter-example to a natural extension of the Penrose Inequality
to charged black holes.
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Week 8
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| A.Ashtekar
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Dynamical Horizons and their properties
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Monday July 21, 16.00
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Abstract:
A detailed description of how black holes grow in full, non-linear
general relativity is presented. The starting point is the notion
of dynamical horizons. Expressions of fluxes of energy and
angular momentum carried by gravitational waves across these
horizons are obtained. Fluxes are local and the energy flux is
positive. Change in the horizon area is related to these fluxes.
The flux formulae also give rise to balance laws analogous to the
ones obtained by Bondi, Sachs and others at null infinity and
provide generalizations of the first and second laws of black hole
mechanics. The framework has a potential application to the proof of
a stronger version of the Penrose inequality where the apparent
horizon area is related to the future limit of Bondi mass.
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| P.Tod
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Some geometrical inequalities implied by the Penrose Equality
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Tuesday July 22, 16.00
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Abstract:
I shall review a number of special situations in which the truth of the
Penrose inequality would imply a geometrical inequality, which one can
then seek to prove or disprove. The special situations include the
Robinson-Trautman metrics, a 'black-hole-in-a-box', the Gibbons-Penrose
construction of trapped surfaces and an example due to Robinson and
Winicour.
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| F.Finster
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Curvature estimates in asymptotically flat Lorentzian manifolds
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Wednesday July 23, 16.00
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Abstract:
The PET implies that an asymptotically flat space-time of zero ADM
energy is flat. It is therefore natural to expect that if the ADM
energy is small, space-time should be "nearly flat". We report on
curvature estimates which make this statement precise and bound the
Riemann tensor in terms of the ADM energy. The proofs are based on
spinors. This is joint work with H. Bray, I. Kath, and M. Kraus.
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| P.Michor
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Riemannian geometries on spaces of plane curves |
Thursday July 24, 11.00
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Abstract: The usual $H0$-metric on the space of embeddings of the circle
to the plane is invariant under reparameterazations and thus it induces a
Riemannian metric on the quotient space, the `space of shapes'. However, the
Riemannian distance between any two shapes is (surprisingly) 0.
A better metric is
$$G^A_c(h,k)= \int_{S1}(1+A\kappa(c)2)|c'|d\theta.$$
Its geodesic equation is a horrible nonlinear PDE, but some nice properties
can be deduced. Its sectional curvature is positive in a central part of the
space (for curves whose curvature stays below some bound).
Many questions remain open. |
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| L.Andersson
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The rescaled Hamiltonian and the sigma constant
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Thursday July 24, 16.00
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Abstract:
The rescaled Hamiltonian $H= |\tau|^3 Vol(M,g)$ is a monotone
quantity along the Einstein flow. In case $M$ is of hyperbolic type, the
lower bound for $H$ is related to the $\sigma$-constant of $M$, resulting in
a "positive mass" conjecture for spatially compact spacetimes. I will review
work relating to this by Fischer and Moncrief, Anderson, and myself.
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| J.Lewandowski
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Non-expanding horizons, isolated horizons and the Killing horizons
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Friday July 25, 15.00
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Abstract:
We will review results concerning the geometry of non-expanding
null surfaces which admit compact, spacelike slices called non-expanding
horizons. Local conditions upon which the geometry of a non-expanding
horizon coincides with that of the Kerr event horizon will be discussed.
In particular, the local version of the uniqueness theorem which holds
for the extremal Kerr horizon will be presented.
Invariants of the non-expanding horizon geometry are useful to formulate
conditions for the existence of a Killing field in a space-time
neighborhood.
An unexpected transformation which maps solutions to the conditions
into solutions of the Kundt null constraints will be presented.
A consequence is an example of an exact vacuum solution foliated by the
Killing horizons constructed from the geometry of the extremal Kerr
event horizon. This is based on joint works with A.Ashtekar, C.Beetle, and T.Pawlowski.
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