  We introduce a family of functionals on submanifolds of Cartan-Hadamard
manifolds that generalize the Colding-Minicozzi entropy of submanifolds of
Euclidean space. We show that 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.
