I will discuss a family of functionals defined on submanifolds of Cartan-Hadamard manifolds that generalize the Colding-Minicozzi entropy of submanifolds of Euclidean space. These quantities are monotone under mean curvature flow under natural conditions. As a consequence, we obtain sharp lower bounds on them for certain closed hypersurfaces and observe a novel rigidity phenomenon. 

This is joint work with A. Bhattacharya.