The uniqueness of the Enneper surfaces and Chern-Ricci functions on minimal surfaces

TL;DR

This paper introduces Chern-Ricci functions on negatively curved minimal surfaces, characterizes Enneper's surface via constant first Chern-Ricci function, and explores the moduli space of surfaces with constant second Chern-Ricci function.

Contribution

It constructs new harmonic functions on minimal surfaces and uniquely characterizes Enneper's surface through the first Chern-Ricci function.

Findings

Enneper's surface is uniquely identified by constant first Chern-Ricci function.

The moduli space of minimal surfaces with constant second Chern-Ricci function includes catenoids, helicoids, and their associate families.

Abstract

We construct the first and second Chern-Ricci functions on negatively curved minimal surfaces in ${\mathbb{R}}^{3}$ using Gauss curvature and angle functions, and establish that they become harmonic functions on the minimal surfaces. We prove that a minimal surface has constant first Chern-Ricci function if and only if it is Enneper's surface. We explicitly determine the moduli space of minimal surfaces having constant second Chern-Ricci function, which contains catenoids, helicoids, and their associate families.