Interaction of Mean Curvature Flow and a Diffusion Equation

We consider a geometric problem consisting of an evolution equation for a closed hypersurface coupled to a parabolic equation on this evolving surface. More precisely, the evolution of the hypersurface is determined by a scaled mean curvature flow that depends on a quantity defined on the surface via a diffusion equation. This system arises as a gradient flow of a simple energy functional. Assuming suitable parabolicity conditions, we derive short-time existence for the system. The proof is based on linearization and a contraction argument. For this, we parameterize the hypersurface via a height function and thus the system, originally defined on an evolving surface, can be transformed onto a fixed reference surface. The result is formulated in a classical sense, holds for the case of embedded and immersed hypersurfaces alike and provides an existence time independent of small changes in the initial surface. Afterwards, several properties of the solution are analyzed. Emphasis is placed on to what extent the surface in our setting evolves the same as for the usual mean curvature flow. To this end, we show that the surface area is stricly decreasing but give an example of a surface that exists for infinite times nevertheless. Moreover, mean convexity is conserved whereas convexity is not. Finally, we construct an embedded hypersurface that develops a self-intersection in the course of time