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Description:
The evolution of hypersurfaces in the direction of the unit normal with speed equal to the reciprocal of the mean curvature is called inverse mean curvature flow (IMCF). In the case of closed hypersurfaces this flow is well studied. One of the classical results goes back to Gerhardt (see also Urbas). He proved long-time existence and convergence to a round sphere for star-shaped initial data with strictly positive mean curvature. A more recent result with a striking application to theoretical physics is due to Huisken and Ilmanen. They defined weak solutions of IMCF and proved existence and uniqueness of such solutions. This was one of the main tools in their proof of the Riemannian Penrose inequality which gives an estimate for the mass in general relativity. In the current work we will investigate IMCF in the case where the hypersurfaces possess a boundary and move along, but stay perpendicular to, a fixed supporting hypersurface. The work is organized as follows: We will use Chapter 1 to give a more detailed overview about geometric evolution equa- tions in general and about IMCF for closed hypersurfaces in particular. Furthermore, we will specify our setup for hypersurfaces with boundary. The first question which we have to answer is whether or not this flow has a solution for a small time. This short-time existence result is obtained in Chapter 2, Theorem 2.12 by writing the hypersurface as a graph over the initial hypersurface and reducing the equations to a scalar parabolic Neumann problem. This approach was also used by Stahl for hypersurfaces with boundary evolving under mean curvature flow. The counter example of a half-torus evolving on a plane shows that long-time existence cannot be expected in general. However, in the case where the supporting hypersurface is a convex cone and the initial hypersurface is star-shaped and has strictly positive mean curvature, we are able to prove long-time existence and convergence to a spherical cap. This work is carried out in Chapter 3. The main result is Theorem ...