SPACELIKE CAPILLARY SURFACES IN THE LORENTZ-MINKOWSKI SPACE

Abstract

For a compact spacelike constant mean curvature surface with nonempty boundary in the three-dimensional Lorentz-Minkowski space, we introduce a rotation index of the lines of curvature at the boundary umbilical point, which was developed by Choe ['Sufficient conditions for constant mean curvature surfaces to be round', Math. Ann. 323(1) (2002), 143-156]. Using the concept of the rotation index at the interior and boundary umbilical points and applying the Poincare-Hopf index formula, we prove that a compact immersed spacelike disk type capillary surface with less than four vertices in a domain of L(3) bounded by (spacelike or timelike) totally umbilical surfaces is part of a (spacelike) plane or a hyperbolic plane. Moreover, we prove that the only immersed spacelike disk type capillary surface inside a de Sitter surface in L(3) is part of (spacelike) plane or a hyperbolic plane.