Renormalized Area of Hypersurfaces in Hyperbolic Spaces

TL;DR

This paper derives formulas for the renormalized area of asymptotically minimal hypersurfaces in higher-dimensional hyperbolic spaces, extending previous results from surfaces in three dimensions to more general settings.

Contribution

It generalizes existing formulas for the renormalized area from minimal surfaces in hyperbolic 3-space to higher dimensions and non-minimal hypersurfaces, including Poincaré-Einstein spaces.

Findings

Derived explicit formulas for renormalized area in higher dimensions

Extended the relation between renormalized area and Willmore energy to non-minimal cases

Generalized results to hypersurfaces of arbitrary codimension in Poincaré-Einstein spaces

Abstract

We employ Chen's conformal invariant quantity [8, Theorem 1] in combination with the Chern-Gauss-Bonnet formulas to obtain expressions for the renormalized area of asymptotically minimal hypersurfaces in the $(2n+1)$-dimensional hyperbolic space $\mathbb{H}^{2n+1}$, $n=1,2$. Our results extend Alexakis and Mazzeo's formula for the renormalized area for surfaces in $\mathbb{H}^3$ [1, Proposition 3.1] as well as their relation between the renormalized area of minimal surfaces of $\mathbb{H}^3$ and the Willmore energy of their doubles in $\mathbb{R}^3$ [1, Proposition 8.1] to the non-minimal case and to the higher dimensional case $n=2$. Moreover, we also generalize our results by considering hypersurfaces in $(2n+1)$-dimensional Poincar\'e-Einstein spaces and even-dimensional submanifolds of arbitrary codimension.