Optimal isoperimetric inequalities for complete proper minimal submanifolds in hyperbolic space

TL;DR

This paper establishes optimal isoperimetric inequalities for minimal submanifolds in hyperbolic space, linking Euclidean and hyperbolic volumes, and introduces the M"{o}bius volume to derive new geometric bounds.

Contribution

It proves new optimal isoperimetric inequalities for minimal submanifolds in hyperbolic space, extending previous Euclidean results and introducing the M"{o}bius volume concept.

Findings

Proved an optimal linear isoperimetric inequality in hyperbolic space.

Established a sharp lower bound for Euclidean volume of minimal submanifolds.

Introduced M"{o}bius volume to derive isoperimetric inequalities.

Abstract

Let $\Sigma$ be a $k$-dimensional complete proper minimal submanifold in the Poincar\'{e} ball model $B^n$ of hyperbolic geometry. If we consider $\Sigma$ as a subset of the unit ball $B^n$ in Euclidean space, we can measure the Euclidean volumes of the given minimal submanifold $\Sigma$ and the ideal boundary $\partial_\infty \Sigma$, say $\rvol(\Sigma)$ and $\rvol(\partial_\infty \Sigma)$, respectively. Using this concept, we prove an optimal linear isoperimetric inequality. We also prove that if $\rvol(\partial_\infty \Sigma) \geq \rvol(\mathbb{S}^{k-1})$, then $\Sigma$ satisfies the classical isoperimetric inequality. By proving the monotonicity theorem for such $\Sigma$, we further obtain a sharp lower bound for the Euclidean volume $\rvol(\Sigma)$, which is an extension of Fraser and Schoen's recent result \cite{FS} to hyperbolic space. Moreover we introduce the M\"{o}bius volume of $\Sigma$ in $B^n$ to prove an isoperimetric inequality via the M\"{o}bius volume for $\Sigma$.