Area bounds for minimal surfaces in geodesic ball of hyperbolic space

TL;DR

This paper establishes lower bounds for the area of minimal submanifolds in hyperbolic space geodesic balls, generalizing classical Euclidean results to hyperbolic geometry.

Contribution

It extends area bound results for minimal surfaces from Euclidean to hyperbolic space, considering submanifolds passing through the center of geodesic balls.

Findings

Minimal submanifolds have area at least that of the totally geodesic submanifold passing through the origin.

The result generalizes Euclidean area bounds to hyperbolic space.

Provides a partial extension of classical minimal surface bounds to non-Euclidean geometry.

Abstract

In hyperbolic space $H^n$ we set a geodesic ball of radius $\rho$. Consider a $k$ dimensional minimal submanifold passing through the origin of the geodesic ball with boundary lies on the boundary of that geodesic ball. We prove that its area is no less than the totally geodesic $k$ dimensional submanifold passing through the origin in that geodesic ball. This is a partial generalization of the corresponding problem in $R^n$.