Colding-Minicozzi entropies in Cartan-Hadamard manifolds

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.