On Geometric Backgrounds of Incompressible Fluid Dynamics on Riemannian Manifolds
The equation of motion of perfect fluids on compact Riemannian manifolds, the Euler equation, has been deduced by Arnold, and the Navier-Stokes equation for incompressible fluids of uniform viscosity was obtained by Taylor.
We explain these works and their geometric background and present and analyze some simple examples.
Also we would like to propose a few problems from the geometric point of view.

Dies ist eine gemeinsame Veranstaltung mit der Topologie (Prof. Dr. E. Vogt)

On singularities of plane curve shortening flows
Several sufficient criteria will be presented under which plane curve shortening flows produce singularities in finite time. Furthermore, the blow up rate of these singularities will be estimated from below.

Closed magnetic geodesics
Magnetic geodesics describe the motion of a charged particle in a magnetic field and correspond to curves with prescribed geodesic curvature.
We give new existence results for closed magnetic geodesics on S².

C^1,α theory for the prescribed mean curvature equation with Dirichlet data
I will discuss regularity of solutions of the prescribed mean curvature equation over a general domain that do not necessarily attain the given boundary data. The work of E.Giusti and others, establishes a very general existence theory of solutions with ”unattained Dirichlet data” by minimizing an appropriately defined functional, which includes information about the boundary data. We can naturally associate to such a solution a current, which inherits a natural minimizing property. The main goal is to show that its support is a C^1,α manifold-with-boundary, with boundary equal to the prescribed boundary data, provided that both the initial domain and the prescribed boundary data are of class C^1,α.

Foliation of asymptotically flat manifolds by surfaces of Willmore type
In this talk I will present aspects of the construction of Willmore type surfaces in asymptotically flat manifolds. The surfaces in question are critical points of the Willmore functional subject to an area constraint. The position vector of these surfaces satisfies a quasi-linear elliptic equation of fourth order. The main result ist that under suitable asymptotic conditions the asymptotic end of an asymptotically flat 3-manifold is foliated by surfaces of Willmore type that converge to Euclidean spheres as the area becomes large.

Boundary regularity for solutions of fully nonlinear, uniformly elliptic equations
The talk will start with a brief introduction to the theory of viscosity solutions. Thereafter we present theorems on interior $W^{2,p}$ and $C^{1, \alpha}$ regularity for viscosity solutions that were proven by L. Caffarelli and A. Swiech. In the main part of the talk we will show how to prove the corresponding boundary regularity.