Rotationally symmetric mean curvature flow with free boundaries

Abstract

In this talk we discuss the setting of rotationally symmetric graphs moving under mean curvature flow with a ninety degree angle condition on fixed rotationally symmetric hypersurfaces. If defined on an annular domain the graphs also enjoy an additional Dirichlet boundary condition. We give sufficient conditions for which the graphs will exist for all times, proved by $C^1$ estimates. By construction of auxiliary functions we prove that they converge to minimal graphs. In some cases we prove that the solution is asymptotically constant. We also give sufficient conditions for which the graphs lose graphicality and develop a curvature singularity at the free boundary. In this talk we will present the details of these arguments.