The second boundary value problem for Gauß curvature flows

We consider the flow of a hypersurface driven by its Gauß curvature. This geometric problem yields a parabolic PDE, a Monge-Ampère equation. We study this equation subject to the second boundary value condition.

In our talk, we will give an introduction to such equations, discuss the obliqueness of the boundary condition, shorttime existence, and a priori estimates required to prove longtime existence. We will study the longtime behavior of these solutions and prove -- depending on the setting -- either convergence to a hypersurface of prescribed Gauß curvature or to a translating solution. For these steps we use only PDE techniques.

If there will be some time left, we will report on results for similar flow equations and boundary conditions.

Part of this work is joint work with Knut Smoczyk.