Freie Universität Berlin -
Fachbereich Mathematik und Informatik
Arnimallee 2-6,
14195 Berlin-Dahlem
(Raum 108/109)
Wintersemester 2004-2005, Dienstag 17.15 Uhr
19.10.2004: Marty Ross
Constant Mean Curvature Surfaces in Asymptotically Flat Spacetimes
Computing Quasilocal Mass
This talk will start with the infimum definition of quasi-local
mass and its more obvious properties. Assuming a natural but possibly
deep conjecture on the nature of the infimum, reduces the calculation of
quasi-local mass to solving an elliptic boundary value problem of Ricci
type. I will describe what little is known about this system, and recent
attempts to numerically compute solutions in the static axially symmetric
case (joint work with Marsha Weaver).
Dirac harmonic maps
Motivated by the non-linear supersymmetric sigma model of quantum field
theory, we introduce and study a variational problem coupling a non-linear
scalar field, taking its values in a Riemannian manifold, and a spinor
field, taking its values in the pull-back tangent bundle of that manifold.
The resulting Euler Lagrange equations couple the harmonic map tension
field with a Dirac equation.
16.11.2004:
Ben Schweizer (Heidelberg/Magdeburg)
On Gamma convergence of SO(2)-invariant functionals modelling
phase transitions
Joint work with Sergio Conti.
The singularly perturbed two-well problem in the theory of
solid-solid phase transitions takes the form
Ie[u] = ∫Ω (1/e) W(∇u) + e
|∇2u|2,
where u:Ω⊂Rn→Rn is the deformation, and W vanishes
for all matrices in K=SO(n)A ∪ SO(n)B.
We focus on the case n=2 and derive, by means of Gamma convergence,
a sharp-interface limit for Ie. The proof is based on a rigidity estimate
for low-energy functions. Our rigidity argument also gives
an optimal two-well Liouville estimate: if ∇u has a small BV
norm (compared to the diameter of the domain), then, in the L1 sense,
either the distance of ∇u from SO(2)A or the one from SO(2)B
is controlled by the distance of ∇u from K. This implies that
the oscillation of ∇u in weak- L1 is controlled by the L1
norm of the distance of ∇u to K.
23.11.2004:
Michel Grüneberg (Albert Einstein Institut)
A geometric convergence proof
for the Yamabe flow
In this talk we consider a geometric evolution equation which is derived from the
total scalar curvature functional on the space of Riemannian metrics on a closed
manifold. Specifically, we study the Yamabe flow, which was originally introduced by
R. Hamilton shortly after the Ricci flow as an alternative approach to solving the
Yamabe problem on manifolds of positive conformal Yamabe invariant. This flow is the
negative L2 gradient flow for the (normalized) total scalar curvature functional
when
restricted to a conformal class. As such, it can be viewed as a natural
geometric deformation of a Riemannian metric to a conformal metric of
constant scalar curvature. Therefore the convergence question for this flow
constitutes the "parabolic version" of Yamabe's problem.
The goal of this talk is to outline a proof of the general convergence
result for this flow on 3-manifolds, given arbitrary initial metrics. The main idea is
to use the local Weil-Petersen geometry near the submanifold of constant curvature
metrics in the standard conformal class of the n-sphere to describe the evolution of
large solutions to the flow. We then show how the Riemannian Pohozaev Identity and the
Riemannian Positive Mass Theorem can be used to rule out that these constant curvature
metrics become arbitrarily concentrated, which suffices to imply convergence of the
flow.
30.11.2004:
Maria Athanassenas (Monash University)
Capillary surfaces in domains with corners
We shall present an overview of results on boundary behaviour for capillary
surfaces in tubes whose cross-sections are domains with corners.
In particular, we shall discuss the Concus-Finn conjecture, and construct an
example of a surface in the gravity-free case, which supports the conjecture.
7.12.2004:
Robert Finn (Stanford)
The contact angle in capillarity
In 1805, Thomas Young gave a reasoning to support the view that
the contact angle at a solid/liquid/gas interface is a physical constant
depending only on the materials, and in no other way on the particular
configuration considered. That conclusion has become one of the
underpinnings of both classical and modern capillarity theory. In this talk,
I intend to raise some questions about it, and offer partial answers.
Foliations of asymptotically flat 3-manifolds by 2-surfaces of prescribed
mean curvature
We construct 2-surfaces of prescribed mean
curvature in 3-manifolds carrying asymptotically flat initial
data for an isolated gravitating system with rather general
decay conditions. The surfaces in question form a regular
foliation of the asymptotic region of such a manifold. We
recover physically relevant data, especially the ADM-momentum,
from the geometry of the foliation.
For a given set of data (M,g,K), with a three dimensional
manifold M, its Riemannian metric g, and the second
fundamental form K in the surrounding four dimensional Lorentz
space time manifold, the equation we solve is H+P=const or
H-P=const. Here H is the mean curvature, and P = tr K is
the 2-trace of K along the solution surface. This is a
degenerate elliptic equation for the position of the surface. It
prescribes the mean curvature anisotropically, since P depends
on the direction of the normal.
14.1.2005:
FRIDAY, 11.15 Uhr, Raum 108/109
Joe Grotowski (City University of New York)
2-dimensional Harmonic map heat flow vs. 4-dimensional
Yang-Mills heat flow
Curvature estimates for F-stable hypersurfaces
In this talk we will consider immersed hypersurfaces in
euclidean (n+1)-space which are stable with respect to an elliptic
parametric functional with integrand F=F(N) depending on normal directions
only. We will present integral and pointwise curvature estimates provided
that F is close to the area integrand, extending the classical estimates
of Schoen-Simon-Yau for stable minimal hypersurfaces as well as Simon's
estimate for F-minimizing hypersurfaces. As a crucial point we will derive
a generalized Simons inequality for the laplacian of a weighted second
fundamental form with respect to an abstract metric associated with F.
Finally, we will briefly discuss the corresponding non-parametric problem.
Positivity of mass for asymptotically AdS spacetimes
I will discuss a proof of positivity of mass for asymptotically
anti-de Sitter (asymptotically AdS) spacetimes. The proof does not make
use of a spin assumption. A central idea in the proof is to study the
brane action L(Σ) = A(Σ) - n V(Σ).
1.2.2005:
Joachim Lohkamp (Münster)
Positive scalar curvature in dimension &ge 8
8.2.2005:
In-Sook Kim
On the Invariance of Domain under
Countably Condensing Vector Fields
A profound topological theorem on the invariance of the
n-dimensional domain was first proved by Brouwer.
Schauder showed that it can be carried over infinite
dimensional spaces.
In this talk, we give an invariance of domain theorem for
countably condensing vector fields, where the notion of
a countably condensing map is due to Väth.
We attempt to prove this in two methods of degree theory
and homotopy theory. A starting point of our investigation
is that there is a compact fundamental set for a countably
condensing map.
15.2.2005:
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