Gauss maps of surfaces in contact space forms. (English) Zbl 1063.53065

Tamura, Michiko

Comment. Math. Univ. St. Pauli 52, No. 2, 117-123 (2003).

In Boll. Unione Mat. Ital., VII. Ser., B 11, No. 2, Suppl., 79–93 (1997; Zbl 0886.53018), A. Sanini studied Gauss maps of surfaces in 3-dimensional Heisenberg group \(H_3\). In particular, he classified all constant mean curvature surfaces in \(H_3\) with vertically harmonic Gauss map. The job of this paper is to generalize Sanini’s result to 3-dimensional contact space forms. Namely, the authors classify constant mean curvature surfaces in 3-dimensional contact space forms with vertically harmonic Gauss maps. The authors prove that if \(M\) is a surface in contact space form \(M^3(c)\), \(c\neq 1\) with constant mean curvature, then the Gauss map of \(M\) is vertically harmonic if and only if \(M\) is a Hopf cylinder of a constant mean curvature. In particular, the only minimal surface in \(M^3(c)\), \(c\neq 1\) with vertically harmonic Gauss map is a Hopf cylinder over a geodesic.

Reviewer: Chiung-Jue Sung (Minhsiung)

MSC:

53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A99	Classical differential geometry
53D15	Almost contact and almost symplectic manifolds

Keywords:

constant mean curvature; Heisenberg group; contact space form; minimal surface; vertically harmonic Gauss map; Hopf cylinder

Citations:

Zbl 0886.53018