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

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.