Conformal Solitons to the Mean Curvature Flow and Minimal Submanifolds

In this note we prove a correspondence, first found by Smoczyk in the hypersurface case, between conformal solitons to the mean curvature flow in an ambient manifold N and minimal submanifolds in a different space N × R. This naturally leads to a new natural stability notion for conformal solitons. We show that this cor- responds to the recent Colding-Minicozzi’s F -stability for codimension one self-shrinkers in euclidean space. In this spirit we can give some classification results for stable conformal solitons, answering two questions of Smoczyk.