A note on isothermic coordinate systems for spacelike surfaces with constant mean curvature in Lorentz-Minkowski space

TL;DR

This work analyzes spacelike surfaces in Lorentz-Minkowski space ${\mathbb{L}}^3$ with constant mean curvature (CMC) by developing isothermic coordinate systems. It proves the existence of such coordinates for non-umbilic CMC surfaces and derives a sinh-Gordon-type equation for the conformal factor $\omega$ that governs the metric, enabling a clean expression of the first and second fundamental forms. Using this framework, the authors obtain global rigidity results: complete non-umbilic CMC surfaces on ${\mathbb{R}}^2$ must be hyperbolic cylinders, and complete CMC surfaces with nonnegative Gaussian curvature must be planes or hyperbolic cylinders. The approach yields simpler proofs of several uniqueness theorems and clarifies the role of the Hopf differential in the isothermic setting, with connections to related geometric structures via the Sym-formula.

Abstract

In this note, we use isothermic coordinate systems to explore global properties of space-like surfaces with constant mean curvature in the Lorentz-Minkowski three-space.

Figures (2)

• Figure 1: hyperbolic cylinder
• Figure 2: pseudosphere

Theorems & Definitions (4)

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