Lifespan theorem for constrained surface diffusion flows
In the analysis of the behaviour of geometric heat flows, the maximum principle has a profound impact. Some natural geometric properties desired of a flow are implied by the maximum principle, such as preservation of convexity of a hypersurface. As is commonly known, the maximum principle is particular to second order partial differential equations. Our work is on fourth order geometric heat flows, and we must resort to different techniques to perform standard analysis. For some fourth order flows, there exists a reasonably large and growing body of literature; for example the Calabi flow, surface diffusion flow, and Willmore flow. For the latter in particular the papers [2, 1, 3] of Kuwert and Sch¨atzle represent a significant achievement and a framework of study which we have attempted to generalise to other flows. Our progress thus far is moderate and in this talk we detail one generalisation to [2]. This is a positive lower bound on the time a smooth solution exists in terms of the concentration of the curvature of the initial surface. We also briefly comment on the applicability of the argument in [1] to surface diffusion flow, which gives smooth convergence to a sphere for initial data with small total tracefree curvature.

 

Isotropic Curvature and the Ricci Flow
In this talk we will discuss a new method to construct invariant curvature cones along the Ricci flow. These cones are defined by inequalities of a curvature function of the frame bundle. Examples include linear combinations of sectional curvatures and positive isotropic curvature. In particular, we will prove in detail that non-negative isotropic curvature is preserved by the flow in dimensions n\geq 4. This was proven by the speaker in his thesis and also by Brendle-Schoen in their proof of the quarter-pinching diffeomorphism sphere theorem.

Rotationally symmetric classical solutions to the Dirichlet problem for Willmore surfaces
The Willmore functional is the integral of the square of the mean curvature over the unknown surface and is to be minimised among all surfaces which obey suitable boundary conditions or, in the case of closed surfaces, constraints of topological or geometrical type. The Willmore equation is the corresponding Euler-Lagrange equation. Quite far reaching results were achieved concerning closed surfaces. Concerning boundary value problems, by far less is known.
In the talk we consider the Willmore equation with Dirichlet boundary conditions for a surface of revolution obtained by rotating the graph of a positive smooth even function. Existence of classical solutions will be discussed.
The lecture is based on joint work with K. Deckelnick (Magdeburg), S. Fröhlich (Free University of Berlin) and H.-Ch. Grunau (Magdeburg).

 

 

Linearly approximatable functions
We say that u, a function from \R^m to R, is linearly $\e$-approximatable at x\in \R^m at scale r>0 if there exists a vector  e in \R^m such that  |u(x+h)-u(x)-<e,h>|<\e r whenever |h|<r. This notion of linear approximability generalises that of being continuously differentiable. It occurs, for instance, in viscosity solutions of some degenerate partial differentiable equations. We establish the Hölder continuity properties of such functions, show also that the family of such functions is meager in the appropriate Hölder space(s), and discuss other properties of linearly approximatable functions.

 

The heat flow and the homology of the loop space We study the moduli space of solutions to the heat equation in a Riemannian manifold with prescribed nondegenerate boundary conditions. These are used to compute the homology of the free loop space of the manifold. In this talk we concentrate on regularity and transversality.

An energy estimate for solutions of the n-dimensional equation with prescribed mean curvature and their removable singularities
We derive an energy estimate for two solutions of the nonparametric equation with prescribed mean curvature in n dimensions. Here we develop ideas by J.C.C.Nitsche for 2-dimensional minimal graphs further to the present situation, where the mean curvature depends monotonically on the solution as well. Identifying possibly singular solutions with those for the corresponding Dirichlet problem, we can remove singularities with vanishing (n-1)-dimensional Hausdorff measure for graphs of constant mean curvature. A similar removability result has been achieved by L. Simon with alternative methods.