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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.

Generic uniqueness of expanders with vanishing relative entropy