Colding Minicozzi Entropy in Hyperbolic Space

TL;DR

This paper defines a new entropy concept for submanifolds in hyperbolic space, exploring its properties such as monotonicity under mean curvature flow and its relation to conformal volume.

Contribution

It introduces a hyperbolic space entropy analogous to Euclidean cases and proves key properties including monotonicity and conformal volume connection.

Findings

Entropy is monotone along mean curvature flow in low dimensions.

Established a link between the entropy and conformal volume.

Provided foundational properties for the new entropy in hyperbolic space.

Abstract

This note introduces a notion of entropy for submanifolds of hyperbolic space analogous to the one introduced by Colding and Minicozzi for submanifolds of Euclidean space. Several properties are proved for this quantity including monotonicity along mean curvature flow in low dimensions and a connection with the conformal volume.