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Solutions of the Björling problem for timelike surfaces in the Lorentz-Minkowski space

We give a number of new examples of timelike minimal surfaces in the Lorentz–Minkowski space. Our method consists of solving the Björling problem by prescribing a circle or a helix as the core curve α and rotating with constant angular speed the unit normal vector field in the normal plane to α. As particular cases, we exhibit new examples of timelike minimal surfaces invariant by a uniparametric group of helicoidal motions. | Turk J Math (2018) 42: 2186 – 2201 © TÜBİTAK doi:10.3906/mat-1801-93 Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Solutions of the Björling problem for timelike surfaces in the Lorentz-Minkowski space Seher KAYA1 ,, Rafael LÓPEZ2,∗, Department of Mathematics, Faculty of Science, Ankara University, Ankara, Turkey 2 Department of Geometry and Topology, Institute of Mathematics (IEMath-GR), University of Granada, Granada, Spain 1 Received: 29.01.2018 • Accepted/Published Online: 01.06.2018 • Final Version: 27.09.2018 Abstract: We give a number of new examples of timelike minimal surfaces in the Lorentz–Minkowski space. Our method consists of solving the Björling problem by prescribing a circle or a helix as the core curve α and rotating with constant angular speed the unit normal vector field in the normal plane to α . As particular cases, we exhibit new examples of timelike minimal surfaces invariant by a uniparametric group of helicoidal motions. Key words: Timelike minimal surface, Björling problem, circle, helix 1. Introduction The Björling problem in Euclidean space asks for the existence and uniqueness of a minimal surface (a surface with zero mean curvature everywhere) that contains a given real analytic curve and a prescribed analytic unit normal along this curve. The solution to the Björling problem was obtained by Schwarz [11]. When the ambient space is the 3 -dimensional Lorentz–Minkowski space L3 , the Björling problem was solved for spacelike surfaces in [1] and for timelike surfaces in [2]. In the present paper we are interested in the solutions of the Björling problem for timelike surfaces in L3 , which can be formulated as follows. Let α : I ⊂ R → L3 be a regular analytic timelike (resp. spacelike) curve and let V : I → L3 be a given unit analytic spacelike vector field along α such that ⟨α′ , V ⟩ = 0 . The Björling problem consists of determining a timelike minimal surface X : Ω ⊂ R2 → L3 , with I × .