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

Applying the general theory about complete spacelike stationary (i.e. zero mean curvature) surfaces in 4-dimensional Lorentz space ${\mathbb{R}}_1^4$, 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 $\tilde{M}$ (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 $\tilde{M}\to {\mathbb{R}}_1^4$ is algebraic-type. We conjecture that there do not exist non-algebraic examples with $-\int K\mathrm{d}M=4\pi$.