Maximal surfaces in Lorentzian Heisenberg space
The Lorentzian Heisenberg space \(\text{Nil}_1^3(\tau)\) is \(\mathbb{R}^3\) considered with the metric \[ ds_\tau^2=dx^2+dy^2-(\tau(ydx-xdy)+dz)^2. \] Here, \(\tau\) is called the bundle curvature parameter. The main aim of this paper is to study space-like surfaces with zero mean curvature in \(\text{Nil}_1^3(\tau)\). The results are the following. Theorem 3.1. The Gauss map \(g\) of a maximal surface in \(\text{Nil}_1^3(\tau)\) is a harmonic map to \(\mathbb{S}^2\). Proposition 3.1. Let \(\tau\neq 0\). The isothermal maximal immersion \(X=(F,h):\Sigma \to \text{Nil}_1^3(\tau)\) can be described by the Gauss map \(g\) and the bundle curvature \(\tau\) as \[ \begin{aligned} F_w&=\frac{2i}{\tau}\cdot\frac{g_w}{(1+|g|^2)^2}, \quad F_{\overline{w}}=-\frac{2i}{\tau}\cdot\frac{g^2\overline{g}_{\overline{w}}}{(1+|g|^2)^2},\\ h_w&=\frac{2i}{\tau}\cdot\frac{\overline{g}g_w}{(1+|g|^2)^2}-\frac{i\tau}{2}(\overline{F}F_w-F\overline{F}_w). \end{aligned} \] The metric induced on the \(w\)-plane via the immersion \(X\) reads \[ ds^2=\frac{4}{\tau^2}\cdot\frac{(1-|g|^2)^2}{(1+|g|^2)^4}\cdot|g_w|^2|dw|^2. \] Theorem 4.1. There is no complete maximal surface in \(\text{Nil}_1^3(\tau)\) with \(\tau\neq 0\). Theorem 5.1. When the space-like immersion \(X:\Sigma\to\text{Nil}_1^3(\tau)\) has zero mean curvature, its Abresch-Rosenberg differential \(\tilde{\mathcal{P}}\) (the perturbed Hopf differential) is holomorphic on the surface \(\Sigma\). The author observes that there is a duality between maximal surfaces in \(\text{Nil}_1^3(\tau)\) and constant mean curvature surfaces, and, in Section 6, as a concrete duality, he constructs the twin correspondence between constant mean curvature \(-\tau\) graphs in \(\mathbb{R}^3\) and maximal graphs in the Lorentzian Heisenberg space \(\text{Nil}_1^3(\tau)\).
scientific article; zbMATH DE number 5880272
scientific article