Special solutions to a geometric curvature flow of evaporation–condensation

In this paper we study the evolution of special hypersurfaces in Euclidean space evolving according to a geometric curvature flow which is motivated from the phenomenon of evaporation-condensation and which was introduced in Mullins (1957). The first example is a closed, embedded and convex curve which does not converge to a point under the (smooth) flow. This shows that the flow behaves differently from curve shortening flow. Further we show existence of two examples of hypersurfaces which translate under this evolution. Their shape is motivated from translators as they appear in the theory of mean curvature flow: A complete, unbounded graph over a finite time interval and in the higher dimensional case a rotationally symmetric entire graph over R n .