Tohoku Mathematical Journal
2017
December
SECOND SERIES VOL. 69, NO. 4

Tohoku Math. J.
69 (2017), 621-635

Title MINIMAL TIMELIKE SURFACES IN A CERTAIN HOMOGENEOUS LORENTZIAN 3-MANIFOLD

Author Sungwook Lee

(Received March 25, 2015)
Abstract. The 2-parameter family of certain homogeneous Lorentzian 3-manifolds which includes Minkowski 3-space, de Sitter 3-space, and Minkowski motion group is considered. Each homogeneous Lorentzian 3-manifold in the 2-parameter family has a solvable Lie group structure with left invariant metric. A generalized integral representation formula which is the unification of representation formulas for minimal timelike surfaces in those homogeneous Lorentzian 3-manifolds is obtained. The normal Gauß map of minimal timelike surfaces in those homogeneous Lorentzian 3-manifolds and its harmonicity are discussed.

Mathematics Subject Classification. Primary 53A10; Secondary 53C30, 53C42, 53C50.
Key words and phrases. De Sitter space, harmonic map, homogeneous manifold, Lorentz surface, Lorentzian manifold, Minkowski space, minimal surface, solvable Lie group, spacetime, timelike surface.

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