Generic uniqueness of expanders with vanishing relative entropy

Abstract

We define a relative entropy for two self-similarly expanding solutions to mean curvature flow of hypersurfaces, asymptotic to the same cone at infinity. Adapting work of White (Indiana Univ Math J 36(3):567–602, 1987) and using recent results of Bernstein (Asymptotic structure of almost eigenfunctions of drift laplacians on conical ends) and Bernstein-Wang (The space of asymptotically conical self-expanders of mean curvature flow), we show that expanders with vanishing relative entropy are unique in a generic sense. This also implies that generically locally entropy minimising expanders are unique.