Generalized Bernstein Theorem for Stable Minimal Plateau Surfaces

TL;DR

This paper proves that stable, orientable minimal Plateau surfaces with quadratic area growth in three-dimensional space must be flat, extending Bernstein's theorem to this class of generalized minimal surfaces.

Contribution

It establishes a generalized Bernstein theorem for stable minimal Plateau surfaces, showing they are flat under certain conditions, which is a novel extension of classical results.

Findings

Stable minimal Plateau surfaces with quadratic area growth are flat.

The theorem extends Bernstein's classical result to a broader class of surfaces.

Provides conditions under which generalized minimal surfaces are necessarily planar.

Abstract

In this paper, we consider a Generalized Bernstein Theorem for a type of generalized minimal surfaces, namely minimal Plateau surfaces. We show that if an orientable minimal Plateau surface is stable and has quadratic area growth in $\mathbb{R}^3 $, then it is flat.