The Björling problem for timelike minimal surfaces in $${\mathbb {R}}^{4}_{1}$$ R 1 4

Abstract

In this paper, we extend and solve the Bjorling problem for timelike surfaces in the ambient space \({\mathbb {R}}^{4}_{1}\). To do this, we define a Gauss map ideally suited to this setting using the split-complex variable and then we obtain a Weierstrass representation formula. We use this to construct new examples and give applications. In particular, we obtain one-parameter families of timelike surfaces in \({\mathbb {R}}^{4}_{1}\) which are solutions of the timelike Bjorling problem. In addition, we establish symmetry principles for the class of minimal timelike surfaces in \({\mathbb {R}}^4_1\).