Oberseminar Analysis, Geometrie und Physik
Freie Universität Berlin - Fachbereich Mathematik und Informatik
Arnimallee 2-6, 14195 Berlin-Dahlem (Raum 031)
Wintersemester 2006-2007, Dienstag 17.00 Uhr

Veranstalter:
-- Prof. Dr. Klaus Ecker (FU Berlin)
-- Prof. Dr. Gerhard Huisken (MPI Gravitationsphysik Golm/FU Berlin)

17.10.2006: Dr. Brian Smith (Freie Universität Berlin)
Blow-up in the Parabolic Scalar Curvature Equation
Consider a manifold foliated by topological 2-spheres. Suppose that the intrinsic geometry of the foliation spheres has been specified. We would like to obtain a manifold of prescribed scalar curvature in a non-conformal way by modifying the metric only in a direction transverse to the foliation spheres. That is, we want to find a function $u$ so that the metric
\[ g=u^2dr^2+h \]
has the desired scalar curvature $R$, where $r$ is the foliating function and $h$ denotes the metric of the foliation spheres. If the area element of $h$ is expanding with increasing $r$ then this gives rise to a parabolic equation for $u$ in which $r$ plays the role of a time variable. It is easily seen by using the maximum principle that in many cases of physical interest the solution blows up at some finite value of $r$, say $r_1$. The purpose of this talk is to discuss a situation in which blow-up occurs in such a way that the metric can nonetheless be continuously extended up to $r_1$, which corresponds to a horizon.

24.10.2006: SFB Kolloquium

31.10.2006: Dr. Felix Schulze (Freie Universität Berlin)
No mass drop for mean curvature flow of mean convex hypersurfaces
A possible evolution of a compact hypersurface in R^(n+1) by mean curvature past singularities is defined via the level set flow. In the case that the initial hypersurface has positive mean curvature, we show that the Brakke flow associated to the level set flow is actually a Brakke flow with equality. We obtain as a consequence that no mass drop can occur along such a flow. As a further application of the techniques used above we give a new variational formulation for mean curvature flow of mean convex hypersurfaces.
7.11.2006: Dr. Mariel Saez (Albert Einstein Institut)
Relaxation of mean curvature flow via the parabolic Ginzburg-Landau equation
I will discuss a method to represent sets evolving under mean curvature flow as nodal sets of the limit of solutions to the parabolic Ginzburg-Landau equation, given by \be \frac{\partial u_\epsilon}{\partial t}-\Delta u_\epsilon +\frac{(\nabla_u W)(u_\epsilon)}{2\epsilon ^2}=0. \label{*}\ee.
More specifically, first I will consider a curve $Gamma$ evolving under curve shortening flow and a potential function $W$ with two minima at $1$ and $-1$. Then I will show that there are solutions $u_\epsilon$ to equation (*) that as $\epsilon \to 0$, satisfy $$\lim_{\epsilon\to 0}u_\epsilon(x,\bar{t})=\left\{\begin{array}{cc}
1& \hbox{for $x$ outside $\Gamma(\lambda,\bar{t})$}\\
0& \hbox{for $x$ on } \Gamma(\lambda,\bar{t})
\\ -1& \hbox{for $x$ inside }\Gamma(\lambda,\bar{t}). \end{array} \right.$$

Then I will show that similar results can be proved for networks of curves evolving under curve shortening flow. I will also discuss some corollaries that can be derived from this representation.

14.11.2006: SFB Kolloquium

21.11.2006: Dr. Miles Simon ( Albert-Ludwigs-Universitaet Freiburg)
Stability of Euclidean space under Ricci flow
We study the Ricci flow and Ricci-DeTurk flow of Riemannian metrics which are $C^0$ and $L^p$ close to the Euclidean metric on $\R^n$. We show that the Ricci-DeTurk flow (using the Euclidean metric as the fixed background metric) of such a metric exists for all time, and that the metric converges to the Euclidean one (in the smooth sense) as time approaches infinity. A similar result is proved for the Ricci flow, although then it is possible that the limit metric is the pull back of the Euclidean metric by a diffeomorphism.
28.11.2006: Dr. Tobias Lamm (Albert Einstein Institut)
Conservation laws for fourth order systems in four dimensions
In the first part of this talk I will briefly review the recent result of Tristan Riviere on the existence of a conservation law for weak solutions of the Euler-Lagrange equation of conformally invariant variational integrals in two dimensions. I will then show how we can adapt these arguments to show the existence of a conservation law for fourth order systems, including biharmonic maps into general target manifolds, in four dimensions. With the help of this conservation law I will then prove the continuity of weak solutions of these systems. If time permits I will also indicate how one can use this conservation law to prove the existence of a unique weak solution of the biharmonic map flow in the energy space.
This is a joint work with Tristan Riviere (ETH Zuerich).

5.12.2006: SFB Kolloquium

12.12.2006: Thierry de Pauw (Katholieke Universiteit Leuven, Belgium)
The Plateau problem: mass vs. size minimization
I will briefly review solutions of the Plateau problem (in every dimension and codimension) contributed simultaneously and independently by H. Fededer and W.H. Fleming on the one hand, and E.R. Reifenberg on the other hand, both in the early 1960s. The Fededer-Fleming approach proves the existence of mass minimizing integral currents with integral coefficients. Mass corresponds to area counting algebraic multiplicities and mass minimizers model some but not all soap films. Size corresponds to area without counting multiplicities but the existence of size minimizing integral currents is known only in some particular cases. Reifenberg's theory deals with size minimizing objects with coefficients in a compact group. I will describe recent results toward the existence of size minimizers, parts of which are common with R. Hardt or D. Pavlica.
19.12.2006: Pierre Bayard (Instituto de Física y Matemáticas, Morelia, Mexico)
Entire spacelike hypersurfaces of constant Gauss curvature.
In this work with Oliver Schnuerer we prove the existence of entire spacelike hypersurfaces of constant Gauss curvature and with prescribed Gauss map image in Minkowski space. We study the logarithmic Gauss curvature flow with given Gauss map image, and prove that these solutions converge to solutions of the equation of constant curvature. Thus the entire spacelike hypersurfaces of constant Gauss curvature that we constructed are dynamically stable.

9.1.2007: SFB Kolloquium

16.1.2007: Christine Guenter (Albert Einstein Institut)
The analysis of linear stability of the Ricci flow along a collapsing solution
I will discuss the analytic aspects of recent joint work with Dan Knopf and Jim Isenberg, in which we demonstrate the linear stability of a flow equivalent to the Ricci flow along various collapsing homogeneous solutions. The linearized operator is not self-adjoint with respect to any weighted $L^2$ function space, and has quadratically unbounded coefficients in its first and zero order terms. Estimates are obtained using the Koiso Bochner formula, and an approximation of the unbounded operator by a sequence of bounded operators.
23.1.2007: Reto Müller (ETH Zürich)
Differential Harnack Inequalities for Parabolic Equations
A differential Harnack type inequality is a pointwise gradient estimate which can be integrated along a path to obtain a classical Harnack inequality as in the standard parabolic PDE theory. Such pointwise estimates were first introduced by Li and Yau in 1986 and have been found for different parabolic problems (in particular geometric flows) since then. In this talk we will concentrate on the Ricci Flow, where we will present some of the relations between Li-Yau type inequalities and monotone integral quantities, such as Perelman's entropy functionals or his L-functional. We will also present some analogous facts for the heat equation on a static manifold, based on an entropy functional found by Lei Ni.
30.1.2007: SFB Kolloquium
6.2.2007: Steffen Fröhlich (Freie Universität Berlin)
On twodimensional nonlinear elleptic systems in Euclidean spaces \mathbb{R}^n
We discuss selected results concerning the geometry and analysis of two-dimensional critical points of various variational problems. On the one hand, this implies the analytical and numerical construction of famous minimal immersions, of special compact immersions of constant mean curvature, and of Willmore-surfaces. On the other hand, we present classical methods to establish local estimates for gradients and curvature of various nonlinear elliptic systems. We infer some results of Bernstein type.

13.2.2007: SFB Kolloquium

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