Variations of Renormalized Volume for Minimal Submanifolds of Poincare-Einstein Manifolds

TL;DR

This paper studies the asymptotic behavior and renormalized volume of minimal submanifolds in Poincare-Einstein manifolds, deriving variation formulas and existence results, with applications to hyperbolic space.

Contribution

It provides new formulas for the first and second variations of renormalized volume for minimal submanifolds of arbitrary codimension in Poincare-Einstein manifolds.

Findings

Derived formulas for volume variations in various codimensions.

Proved existence of asymptotic descriptions over boundary cylinders.

Established an $L^2$-inner product relation in hyperbolic space.

Abstract

We investigate the asymptotic expansion and the renormalized volume of minimal submanifolds, $Y^m$ of arbitrary codimension in Poincare-Einstein manifolds, $M^{n+1}$. In particular, we derive formulae for the first and second variations of renormalized volume for $Y^m \subseteq M^{n+1}$ when $m < n + 1$. We apply our formulae to the codimension $1$ and the $M = \mathbb{H}^{n+1}$ case. Furthermore, we prove the existence of an asymptotic description of our minimal submanifold, $Y$, over the boundary cylinder $\partial Y \times \mathbb{R}^+$, and we further derive an $L^2$-inner-product relationship between $u_2$ and $u_{m+1}$ when $M = \mathbb{H}^{n+1}$. Our results apply to a slightly more general class of manifolds, which are conformally compact with a metric that has an even expansion up to high order near the boundary.