Spacelike capillary surfaces in the Lorentz-Minkowski space

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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 umbilic point, which was developed by Choe Choe. Using the concept of the rotation index at the interior and boundary umbilic points and applying the Poincaré-Hopf index formula, we prove that a compact immersed spacelike disk type capillary surface with less than 4 vertices in a domain of L³ bounded by (spacelike or timelike) totally umbilic surfaces is part of a (spacelike) plane or a hyperbolic plane. Moreover we prove that the only immersed spacelike disk type capillary surface inside de Sitter surface in L³ is part of (spacelike) plane or a hyperbolic plane.