Relative Entropy of Hypersurfaces in Hyperbolic Space

TL;DR

This paper introduces a relative entropy concept for hypersurfaces in hyperbolic space, linking it to renormalized area and establishing a monotonicity formula for mean curvature flows, advancing geometric analysis in hyperbolic geometry.

Contribution

It defines a new relative entropy measure for hypersurfaces in hyperbolic space and connects it to existing renormalized area concepts, providing tools for analyzing mean curvature flows.

Findings

Relates relative entropy to Graham-Witten's renormalized area

Establishes a monotonicity formula for mean curvature flows in hyperbolic space

Provides new insights into geometric properties of hypersurfaces

Abstract

We study a notion of relative entropy for certain hypersurfaces in hyperbolic space. We relate this quantity to the renormalized area introduced by Graham-Witten[RW99]. We also obtain a monotonicity formula for relative entropy applied to mean curvature flows in hyperbolic space.