HYPERBOLIC GAUSS MAPS AND PARALLEL SURFACES IN HYPERBOLIC THREE-SPACE

Abstract

In the differential geometry of surfaces in hyperbolic three-space H, surfaces of constant mean curvature one (CMC-1 surfaces) and surfaces of constant Gaussian curvature zero (flat surfaces) are well studied. (cf. Refs. 1,2,4,6–13, etc.) These surfaces have representation formulas which turn the complex function theory into the efficacious tool. They play the role such as the Weierstrass-Enneper formula in the minimal surface theory of Euclidean space. Galvez, Martinez and Milan (Ref. 3) also derived a representation formula for a certain class of Weingarten surfaces in H, which contains CMC-1 surfaces and flat surfaces. These Weingarten surfaces are the ones satisfying α(H − 1) = βK for some constants α and β, where K denotes the Gaussian curvature and H the mean curvature. In Ref. 5, the author gave a refinement for their representation formula. In these formulas, the hyperbolic Gauss maps (see Section 2 for the definition) play an important role. Moreover, one of the remarkable thing is that this class of Weingarten surfaces is closed under taking the parallel surfaces. The author thinks it rather important to investigate hyperbolic Gauss maps and parallel surfaces themselves, before representation formulas. For this reason, the purpose of this note is to collect some elementary properties of hyperbolic Gauss maps and parallel surfaces.