The prescribed point area estimate for minimal submanifolds in constant curvature

TL;DR

This paper establishes sharp area estimates for minimal submanifolds passing through a point within geodesic balls in hyperbolic space and, in some cases, in spherical space, extending Euclidean results.

Contribution

It provides new sharp area bounds for minimal submanifolds in hyperbolic and spherical spaces, generalizing previous Euclidean estimates.

Findings

Sharp area estimate for minimal submanifolds in hyperbolic space

Extension of estimates to certain spherical cases

Results are analogous to Euclidean estimates by Brendle and Hung

Abstract

We prove a sharp area estimate for minimal submanifolds that pass through a prescribed point in a geodesic ball in hyperbolic space, in any dimension and codimension. In certain cases, we also prove the corresponding estimate in the sphere. Our estimates are analogous to those of Brendle and Hung in the Euclidean setting.