Area and spectrum estimates for stable minimal surfaces

TL;DR

This paper investigates the area growth and spectral properties of complete stable minimal surfaces in three-dimensional manifolds with scalar curvature bounds, providing explicit estimates and confirming classical results in Euclidean space.

Contribution

It offers new direct proofs of area growth rates and establishes explicit spectrum bounds for minimal surfaces in curved ambient spaces.

Findings

Area growth matches Euclidean plane in Euclidean space

Explicit area growth estimates in hyperbolic space

Upper bounds for bottom spectrum based on scalar curvature

Abstract

This note concerns the area growth and bottom spectrum of complete stable minimal surfaces in a three-dimensional manifold with scalar curvature bounded from below. When the ambient manifold is the Euclidean space, by an elementary argument, it is shown directly from the stability inequality that the area of such minimal surfaces grows exactly as the Euclidean plane. Consequently, such minimal surfaces must be at, a well-known result due to Fisher-Colbrie and Schoen as well as do Carmo and Peng. In the case the ambient manifold is the hyperbolic space, explicit area growth estimate is also derived. For the bottom spectrum, upper bound estimates are established in terms of the scalar curvature lower bound of the ambient manifold.