Complete stationary surfaces in $\mathbb{R}^4_1$ with total curvature $-\int K\mathrm{d}M=4\pi$

TL;DR

This paper classifies complete stationary surfaces in 4D Lorentz space with total curvature 4π, showing they are either generalized catenoids or Enneper surfaces, and explores non-orientable cases with curvature bounds.

Contribution

It provides a classification of algebraic stationary surfaces with total curvature 4π and extends the theory to non-orientable cases, proposing a conjecture on the non-existence of certain non-algebraic examples.

Findings

Surfaces with total curvature 4π are congruent to generalized catenoids or Enneper surfaces.

Total Gaussian curvature for non-orientable surfaces is bounded below by 2π(g+3).

Conjecture: no non-algebraic examples with total curvature 4π exist.

Abstract

Applying the general theory about complete spacelike stationary (i.e. zero mean curvature) surfaces in 4-dimensional Lorentz space $\mathbb{R}^4_1$, we classify those regular algebraic ones with total Gaussian curvature $-\int K\mathrm{d}M=4\pi$. Such surfaces must be oriented and be congruent to either the generalized catenoids or the generalized enneper surfaces. For non-orientable stationary surfaces, we consider the Weierstrass representation on the oriented double covering $\widetilde{M}$ (of genus $g$) and generalize Meeks and Oliveira's M\"obius bands. The total Gaussian curvature are shown to be at least $2\pi(g+3)$ when $\widetilde{M}\to\mathbb{R}^4_1$ is algebraic-type. We conjecture that there do not exist non-algebraic examples with $-\int K\mathrm{d}M=4\pi$.