SOME STRUCTURE THEOREMS FOR COMPLETE H-SURFACES IN HYPERBOLIC 3-SPACE H~3

摘要

Let M be a properly embedded connected constant mean curvature surface (nonzero) in hyperbolic 3-space H~3 with boundary a strictly convex curve C. We assume M is complete and C is contained in a geodesic plane P. Let H_+~3 be one of the two half-spaces determined by P. In [NR] it is shown that when M is compact and transverse to P along C, then M is entirely contained in a half-space of H~3 determined by P. Then all the symmetries of C are also symmetries of M; in particular, M is spherical if C is a circle. In Euclidean 3-space, some interesting results on complete noncompact H-surfaces are obtained in [RS. 1 ]. Our main contribution is to extend this work to the hyperbolic case.