A gap theorem for free boundary minimal surfaces in geodesic balls of hyperbolic space and hemisphere

TL;DR

This paper establishes a pinching condition involving geometric quantities to characterize specific minimal surfaces with free boundary in geodesic balls of hyperbolic space and hemisphere, extending understanding of their geometric properties.

Contribution

It introduces a novel pinching condition that uniquely characterizes totally geodesic disks and rotational annuli among free boundary minimal surfaces in these spaces.

Findings

Characterization of totally geodesic disks

Identification of rotational annuli

New geometric inequalities involving second fundamental form and support function

Abstract

In this paper we provide a pinching condition for the characterization of the totally geodesic disk and the rotational annulus among minimal surfaces with free boundary in geodesic balls of three-dimensional hyperbolic space and hemisphere. The pinching condition involves the length of the second fundamental form, the support function of the surface, and a natural potential function in hyperbolic space and hemisphere.