SCREW MOTION SURFACES IN H~2 x R AND S~2 x R

摘要

In this paper we study the geometry of constant mean curvature H-screw motion surfaces in H~2 X R and S~2 X R. We compute the Abresch-Rosenberg holomorphic quadratic differential Q. For instance, if the ambient space is H~2 x R, we derive that, given l, if 0 < 4H~2 < 1, then there exists a complete if—screw motion surface with pitch l immersed in H~2 x R, such that Q ≠ 0. An analogous result holds if the ambient space is S~2 x R. We deduce a general non-parametric formula for the mean curvature H(ρ). When H is constant, we find a first integral and use it to get an explicit two parameter family of complete, embedded, simply connected, minimal screw motion surfaces in H~2 x R with pitch l. If l = 1, each such surface has Gaussian curvature K ≡ -1. We deduce that any two isometric screw motion minimal immersions in H~2 x R or S~2 x R are associate, i.e., the absolute values of their Hopf functions are the same.