Spacelike capillary surfaces in the Lorentz-Minkowski space

TL;DR

This paper classifies compact spacelike capillary surfaces with constant mean curvature in Lorentz-Minkowski space, showing they are parts of planes or hyperbolic planes under certain boundary conditions, using the rotation index method.

Contribution

It introduces a rotation index for boundary umbilic points and applies it to classify spacelike capillary surfaces in Lorentz-Minkowski space, extending previous geometric results.

Findings

Such surfaces are parts of planes or hyperbolic planes.

The classification holds for surfaces with fewer than 4 vertices.

The only spacelike capillary surface in de Sitter space is a plane or hyperbolic plane.

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 umbilic point, which was developed by Choe \cite{Choe}. Using the concept of the rotation index at the interior and boundary umbilic points and applying the Poincar\'{e}-Hopf index formula, we prove that a compact immersed spacelike disk type capillary surface with less than $4$ vertices in a domain of $\Bbb L^3$ 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 $\Bbb L^3$ is part of (spacelike) plane or a hyperbolic plane.