Ricci flow of non-negatively curved 3-manifolds with cone-like singularities
  

Stability of Translating Solutions to Mean Curvature Flow
We prove stability of rotationally symmetric translating solutions to mean curvature flow. For initial data that converge spatially at infinity to such a soliton, we obtain convergence for large times to that soliton without imposing any decay rates.

Exponential Stability for Wave Equations with Indefinite (Non-Dissipative) Damping

We consider the non-linear wave equation , , where the function is allowed to change sign, but has to satisfy . New conditions are presented for this non-dissipative situation with indefinite damping term, under which the linearized system is exponentially stable, and the nonlinear system is globally well-posed in the small.

The inverse mean curvature flow and p-harmonic functions
We consider the level set formulation of the inverse mean curvature flow. We establish a connection to the problem of p-harmonic functions and give a new proof for the existence of weak solutions.

Harmonicity in the presence of complex structures
The problem of defining harmonic maps in the presence of complex structures is discussed. First we follow a proposal of Jost and Yau in the case of a complex structure on the preimage manifold and give an existence and uniqueness result. The discussion of examples will suggest properties of so-called 'hermitian-harmonic maps' quite different from the real harmonic situation. So we propose to pursue a different approach.

The double bubble conjecture in R^n
In three-dimensional Euclidean space, a standard double bubble describes the familiar surface which is formed when two soap bubbles join to enclose two volumes. As Plateau empirically observed, such a surface consists of three spherical caps which meet at 120 degree angles. Analogously, a standard double bubble in n-dimensional Euclidean space is defined as a collection of three (n-1)-dimensional spherical caps which intersect at 120 degree angles along a common (n-2)-dimensional sphere. The double bubble conjecture states that in n-dimensional Euclidean space, a standard double bubble is the surface of minimal area which encloses two volumes. This conjecture has been proven for n=2, n=3, and n=4, but a general proof of the conjecture in higher dimensional spaces has remained elusive. I will describe recent progress toward a solution of the general problem in R^n.

Nonlinear Evolution by Mean Curvature and the Isoperimetric Inequality
Consider a family of smooth compact hypersurfaces in R^(n+1) evolving with normal speed equal to a positive power k of the mean curvature. For k > n-1, smooth solutions to such flows improve a certian 'isoperimetric difference'. If a smooth flow exists until the volume decreases to zero, this proves the isoperimetric inequality for the initial configuration. In general, singularities will develop before the volume goes to zero. To deal with this problem, we develop a weak level-set formulation for such flows and show that the above monotonicity is still valid. This proves the isoperimetric inequality for n < 7. Extending this to complete, simply connected 3- dimensional manifolds with nonpositive sectional curvature, we give a new proof for the Euclidian isoperimetric inequality on such manifolds.

Evolution of an extended Ricci flow system
By extending the Ricci Flow with a scalar function I obtain a new PDE system whose stationary points are static Einstein vacuum metrics. I show how this system can be derived from a variational principle which allows me, following recent ideas of G. Perelman, to deduce a noncollapsing result for the flow. The second major tool I want to present are local interior a priori estimates for solutions. Together these two ingredients imply results on long time existence and on the singularity structure of the flow and further information with regard to asymptotically flat manifolds in General Relativity.

Maximumprinzipien für Flächen mit beliebiger Kodimension
Es werden quadratische Formen konstruiert, die subharmonisch auf Minimalflächen bzw. Flächen mit beschränkter mittlerer Krümmung sind. Dies führt zu Nichtexistenzresultaten für zusammenhängende Minimalflächen beliebiger Kodimension. Ferner wird ein Barriereprinzip für k-kodimensionale Flächen mit (geeignet ) beschränktem mittlerem Krümmungsvektor diskutiert.

New compactness and existence theorems for Ricci flow