A local rigidity theorem for minimal surfaces in Minkowski 3-space of Randers type

Abstract

Let \((\mathbb{R}^3,\widetilde{F}_b)\) be a Minkowski 3-space of Randers type with \(\widetilde{F}_b=\widetilde{\alpha}+\widetilde{\beta}\), where \(\widetilde{\alpha}\) is the Euclidean metric and \(\widetilde{\beta}=bdx^3,0 < b < 1\). We consider minimal surfaces in \((\mathbb{R}^3,\widetilde{F}_b)\) and prove that if a connected surface M in \(\mathbb{R}^3\) is minimal with respect to both the Busemann–Hausdorff volume form and the Holmes–Thompson volume form, then up to a parallel translation of \(\mathbb{R}^3\), M is either a piece of plane or a piece of helicoid which is generated by lines screwing about the x3-axis.