Minimal surfaces and Schwarz lemma

TL;DR

This paper establishes a sharp Schwarz lemma for minimal surfaces represented via the Weierstrass-Enneper parameterization, characterizing when equality occurs and identifying the surface as an affine disk.

Contribution

It introduces a precise Schwarz type inequality for minimal surfaces and characterizes the cases of equality, extending classical results to minimal surface theory.

Findings

Proves a sharp inequality for conformal harmonic maps of the disk to minimal surfaces.

Characterizes equality cases as affine disks with linear parameterizations.

Extends Schwarz lemma concepts to the setting of minimal surface representations.

Abstract

We prove a sharp Schwarz type inequality for the Weierstrass- Enneper representation of the minimal surfaces. It states the following. If $F:\mathbf{D}\to \Sigma$ is a conformal harmonic parameterization of a minimal disk $\Sigma$, where $\mathbf{D}$ is the unit disk and $|\Sigma|=\pi R^2$, then $|F_x(z)|(1-|z|^2)\le R$. If for some $z$ the previous inequality is equality, then the surface is an affine disk, and $F$ is linear up to a M\"obius transformation of the unit disk.