THE GAUSS MAP AND SPACELIKE SURFACES WITH PRESCRIBED MEAN CURVATURE IN MINKOWSKI 3-SPACE

DOI 機関リポジトリ 機関リポジトリ (HANDLE) 被引用文献7件 参考文献6件 オープンアクセス

書誌事項

公開日
1990
資源種別
departmental bulletin paper
DOI
  • 10.2748/tmj/1178227694
公開者
東北大学大学院理学研究科数学専攻

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説明

For an oriented spacelike surface M in Minkowski 3-space \(L^ 3\), the Gauss map G is defined to be a mapping of M into the unit pseudosphere \({\mathbb{H}}\) in \(L^ 3\) assigning to each point p of M the timelike unit normal vector at p translated parallelly to the origin. In this paper, the authors prove a representation formula for spacelike surfaces with prescribed mean curvature in terms of their Gauss maps. More precisely, the following are proved. (1) Arbitrary oriented spacelike surfaces in \(L^ 3\) satisfy a system of first order partial differential equations involving the mean curvature function H and the Gauss map G. (2) The complete integrability condition for this system yields a system of second order partial differential equations identifying the gradient of H and the tension field of G, which simply means that the Gauss map G should be a harmonic mapping if the mean curvature H is constant. (3) Conversely, given a nowhere holomorphic smooth mapping G of a simply connected Riemann surface M into the pseudosphere \({\mathbb{H}}\) satisfying the complete integrability condition for some nonvanishing smooth function H on M, one can construct explicitly a spacelike immersion of M into \(L^ 3\) such that the mean curvature of M is H and the Gauss map of M is given by G. These constitute a Lorentzian counterpart of the Weierstrass-Enneper-Kenmotsu representation formula for surfaces in Euclidean 3-space.

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