Conformal solitons to the mean curvature flow and minimal submanifolds

Abstract

Abstract 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 \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$N\times \textbf {R}$\end{document} . This naturally leads to a new natural stability notion for conformal solitons. We show that this corresponds 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.