Colding–Minicozzi entropies in Cartan–Hadamard manifolds

Abstract We introduce a family of functionals defined on the set of submanifolds of Cartan–Hadamard manifolds which generalize the Colding–Minicozzi entropy of submanifolds of Euclidean space. We show these functionals are monotone under mean curvature flow under natural conditions. As a consequence, we obtain sharp lower bounds on these entropies for certain closed hypersurfaces and observe a novel rigidity phenomenon.