Oberseminar Analysis, Geometrie und Physik
Freie Universität Berlin - Fachbereich Mathematik und Informatik
Arnimallee 2-6, 14195 Berlin-Dahlem (Raum 108/109)
Wintersemester 2004-2005, Dienstag 17.15 Uhr

Veranstalter:
-- Prof. Dr. Klaus Ecker (FU Berlin)
-- Prof. Dr. Gerhard Huisken (MPI Gravitationsphysik Golm/FU Berlin)
-- Julie Clutterbuck (FU Berlin), Gruppe Geometric Analysis.

19.10.2004: Marty Ross
Constant Mean Curvature Surfaces in Asymptotically Flat Spacetimes
26.10.2004: Thomas Friedrich (Humboldt-Universität)
Characteristic torsions of geometric structures
Abstract (PostScript)
02.11.2004: Robert Bartnik (University of Canberra)
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).
09.11.2004: Jürgen Jost (MPI Leipzig)
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.
14.12.2004: Jan Metzger (Albert Einstein Institut)
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
18.1.2005: Sven Winklmann (Duisburg)
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.
25.1.2005: Lars Andersson (Miami)
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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