Colding-Minicozzi Entropies in Cartan-Hadamard Manifolds

TL;DR

This paper introduces a new family of entropy functionals for submanifolds in Cartan-Hadamard manifolds, demonstrating their monotonicity under mean curvature flow and deriving sharp bounds and rigidity results.

Contribution

It generalizes Colding-Minicozzi entropy to Cartan-Hadamard manifolds and establishes their monotonicity and geometric bounds.

Findings

Monotonicity of the new entropy functionals under mean curvature flow

Sharp lower bounds for entropies of certain closed hypersurfaces

Identification of a rigidity phenomenon related to entropy bounds

Abstract

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.