Complete stationary surfaces in R^4_1 with total Gaussian curvature 6\pi

TL;DR

This paper classifies complete stationary surfaces in 4D Lorentz space with total Gaussian curvature 6π, revealing their topological type as Möbius strips and constructing new examples with singular ends.

Contribution

It extends previous classification results to total curvature 6π, identifying topological constraints and providing new explicit examples.

Findings

Surfaces with total curvature 6π are topologically Möbius strips.

Existence of new examples with a single good singular end.

Topological and geometric properties of these surfaces are characterized.

Abstract

In a previous paper we classified complete stationary surfaces (i.e. spacelike surfaces with zero mean curvature) in 4-dimensional Lorentz space $\mathbb{R}^4_1$ which are algebraic and with total Gaussian curvature $-\int K\mathrm{d}M=4\pi$. Here we go on with the study of such surfaces with $-\int K\mathrm{d}M=6\pi$. It is shown in this paper that the topological type of such a surface must be a M\"obius strip. On the other hand, new examples with a single good singular end are shown to exist.