The harmonic mean curvature flow of a 2-dimensional hypersurface

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Natasa Sesum, Columbia University
Fine Hall 314

The harmonic mean curvature flow is the flow that moves a hypersurface embedded in $R^3$ by the speed given by a ratio of the Gauss and the mean curvature of the given surface in the direction of its normal. It is a fully nonlinear, weakly parabolic equation, degenerate at the points at which our hypersurface changes its convexity and fast diffusion when the mean curvature tends to zero. We prove a short time existence of such a flow in a nonconvex case. We also prove that if the mean curvature does not go to zero, the flow becomes strictly convex at some time and shrinks to a round point. This is an example of a curvature flow in higher dimension (besides the curve shortening flow in a plane which has been known for a long time) that exhibits a nice shrinking property in a spherical manner in a finite time, even if we start evolving nonconvex hypersurfaces. In that sense this flow behaves better that the mean curvature flow that in the most nonconvex cases develops singularities before shrinking to a point.