Oberseminar Analysis, Geometrie und Physik
Freie Universität Berlin - Fachbereich Mathematik und Informatik
Arnimallee 6, 14195 Berlin-Dahlem (Raum 031)
Sommersemester 2011, Dienstag 17.00 Uhr

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

Rigidity estimates for mean curvature flow and Ricci flow
19.04.2011: SFB Kolloquium

26.04.2011: Klaus Ecker (FU Berlin)
Partial regularity at the first singular time for hypersurfaces evolving by mean curvature
We consider smooth, properly embedded hypersurfaces evolving by mean curvature in some open subset of $\mathbb{R}^{n+1}$ on a time interval $(0, t_0)$. We prove that $p$ - integrability with $p\ge 2$ of the second fundamental form of these hypersurfaces in some space-time region $B_R(y)\times (0, t_0)$ implies that the $\mathcal{H}^{n+2-p}$- measure of the first singular set vanishes inside $B_R(y)$. By work of Ilmanen, this integrability condition is satisfied for $p=2$ and $n=2$ if the initial surface has finite genus. Thus the first singular set has zero $\mathcal{H}^2$-measure in this case. This is the conclusion of Brakke's main regularity theorem for the special case of surfaces, but derived without having to impose the area continuity and unit density hypothesis. Recent results of John Head imply that for closed, two-convex hypersurfaces our integrability criterion holds with exponent $p=n+1-\alpha$ for all small $\alpha >0$. Therefore, the first singular set is at most one-dimensional in this situation.
10.05.2011: SFB Kolloquium

24.05.2011: Brett Kotschwar (AEI)
Some unique continuation results for the Ricci and mean curvature flows
We discuss some applications of a general backwards uniqueness result for weakly parabolic systems with certain "geometric" degeneracies. The first is a backwards uniqueness result for the Ricci flow: two solutions that agree at some non-initial time must agree at at all previous times. In particular, a solution cannot acquire a novel isometry within its smooth lifetime nor become Einstein unless it is identically so. The second is a proof that the holonomy group of a solution to Ricci flow cannot spontaneously contract in finite time; thus a solution is locally reducible or Kaehler only if it is so at previous times. We will also discuss the counterparts of these results for the mean curvature flow, and a variant of the method for weakly-elliptic systems which yields, for example, an alternative path to unique continuation results for Ricci solitons (avoiding analyticity).
31.05.2011: SFB Kolloquium

7.06.2011: Felix Schulze (FU)
On short time existence of the network flow.
I will report on joint work with T. Ilmanen and A. Neves on how to prove the existence of an embedded, regular network moving by curve shortening flow in the plane, starting from a non-regular initial network. Here a regular network consists of smooth, embedded line-segments such that at each endpoint, if not infinity, there are three arcs meeting under a 120 degree angle. In the non-regular case we allow that an arbitrary number of line segments meet at an endpoint, without an angle condition. The proof relies on gluing in appropriately scaled self-similarly expanding solutions and a new monotonicity formula, together with a local regularity result for such evolving networks. This short time existence result also has applications in extending such a flow of networks through singularities.
Expanding solitons with non-negative curvature operator coming out of cones
Joint work with: Felix Schulze Abstract: We show that a Ricci flow of any complete Riemannian manifold without boundary with bounded non-negative curvature operator and non-zero asymptotic volume ratio exists for all time and has constant asymptotic volume ratio. We show that there is a limit solution, obtained by scaling down this solution at a fixed point in space, which is an expanding soliton coming out of the asymptotic cone at infinity.
21.06.2011: SFB Kolloquium

From L^2 estimates to Harnack inequalities and two sided Heat kernel bounds
We consider operators of the form \mathcal L = - L - V , where L is an elliptic operator and V is a singular potential, defined on a smooth bounded domain \Omega \subset R^n with Dirichlet boundary conditions. We allow the boundary of \Omega to be made of various pieces of different codimension. We assume that \mathcal L has a generalized first eigenfunction of which we know two sided estimates. Under these assumptions we prove optimal Sobolev inequalities for the operator \mathcal L and derive a parabolic Harnack inequality up to the boundary as well as sharp heat kernel estimates.
Joint work with: S. Filippas and L. Moschini
12.07.2011: SFB Kolloquium

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