Surfaces of annulus type with constant mean curvature in Lorentz-Minkowski space

TL;DR

This paper solves the Plateau problem for spacelike surfaces with constant mean curvature in Lorentz-Minkowski space, characterizing rotationally symmetric solutions with specific boundary conditions and singularities.

Contribution

It provides a classification of spacelike constant mean curvature surfaces bounded by two circles, identifying rotational symmetry as the key feature.

Findings

Rotational symmetric surfaces are the only compact solutions with given boundary conditions.

Such surfaces either solve the exterior Dirichlet problem or have a conical singularity.

The paper characterizes all solutions with these properties.

Abstract

In this paper we solve the Plateau problem for spacelike surfaces with constant mean curvature in Lorentz-Minkowski three-space $\l^3$ and spanning two circular (axially symmetric) contours in parallel planes. We prove that rotational symmetric surfaces are the only compact spacelike surfaces in $\l^3$ of constant mean curvature bounded by two concentric circles in parallel planes. As conclusion, we characterize spacelike surfaces of revolution with constant mean curvature as the only that either i) are the solutions of the exterior Dirichlet problem for constant boundary data or ii) have an isolated conical-type singularity.