Chern-Ricci harmonic functions on zero mean curvature surfaces in the three-dimensional Lorentz-Minkowski space and the rigidity of Enneper's surface

Abstract

In this paper, we introduce Chern-Ricci functions on spacelike maximal surfaces and timelike minimal surfaces in L3, and we demonstrate that they are harmonic. Thereafter, we obtain a rigidity (uniqueness) result for classical examples of surfaces with zero mean curvature in L3 using the constancy of the Chern-Ricci functions. Particularly, we prove that Enneper's surface is the only surface that possesses zero mean curvature with a constant first Chern-Ricci function. We also prove that surfaces having zero mean curvature with a constant second Chern-Ricci function are contained in the moduli space of catenoids, helicoids, and their associated families up to homotheties.