[Paper Review] Surfaces of annulus type with constant mean curvature in Lorentz-Minkowski space

Rafael López|ArXiv.org|Jan 12, 2005

Geometric Analysis and Curvature Flows20 references5 citations

TL;DR

This paper establishes that rotational symmetric spacelike surfaces with constant mean curvature (CMC) are the only compact CMC surfaces in Lorentz-Minkowski space bounded by two concentric circles in parallel planes. Using the tangency principle and flux comparison, it proves uniqueness under Dirichlet or conical singularity conditions, resolving the Plateau problem for annular CMC surfaces in L³.

ABSTRACT

In this paper we solve the Plateau problem for spacelike surfaces with constant mean curvature in Lorentz-Minkowski three-space $ł^3$ and spanning two circular (axially symmetric) contours in parallel planes. We prove that rotational symmetric surfaces are the only compact spacelike surfaces in $ł^3$ 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.

Motivation & Objective

• To solve the Plateau problem for spacelike surfaces with constant mean curvature (CMC) in Lorentz-Minkowski 3-space (L³), spanning two concentric circular boundaries in parallel planes.
• To determine the necessary and sufficient conditions for the existence of such CMC surfaces, particularly focusing on rotational symmetry.
• To prove that rotational symmetric CMC surfaces are the only compact spacelike CMC surfaces bounded by two concentric circles in parallel planes.
• To characterize CMC surfaces of revolution as the only ones that either solve the exterior Dirichlet problem with constant boundary data or exhibit an isolated conical-type singularity.
• To establish uniqueness of CMC solutions via the tangency principle and flux comparison, under symmetry and boundary conditions.

Proposed method

• Formulates the CMC equation in Lorentz-Minkowski space using a rotational graph parametrization: $ X(t,\theta) = (t\cos\theta, t\sin\theta, f(t)) $, with $ f(r) = a $, $ f(R) = b $, $ r < R $.
• Applies the tangency principle for CMC surfaces: if two CMC surfaces with same $ H $ touch tangentially at an interior or boundary point, they must coincide locally.
• Uses flux comparison along the boundary curve $ \Gamma(r,a) $: equates $ \int_{\Gamma(r,a)} \langle \nu, \mathbf{e}_3 \rangle \, ds $ for two surfaces to derive contradiction if they do not coincide.
• Employs a one-parameter family of translated surfaces $ \Sigma_1(t) = \Sigma_1 + t\mathbf{e}_3 $ to analyze contact behavior and apply maximum principle arguments.
• Applies analytic continuation via the analyticity of solutions to elliptic PDEs to conclude global coincidence of surfaces when they touch.
• Extends the argument to entire graphs with isolated conical singularities by reversing the comparison and proving mutual dominance.

Experimental results

Research questions

• RQ1 Under what conditions does a spacelike CMC surface of annulus type exist in L³ spanning two concentric circles in parallel planes?
• RQ2 Are rotational symmetric CMC surfaces the only compact spacelike CMC surfaces bounded by two concentric circles in parallel planes in L³?
• RQ3 Can the exterior Dirichlet problem for the CMC equation in L³ be solved uniquely when the boundary data are constant and the surface is spacelike?
• RQ4 What characterizes CMC surfaces with an isolated conical-type singularity in terms of uniqueness and boundary behavior?
• RQ5 Does the tangency principle imply uniqueness of CMC surfaces in L³ when they share boundary data or flux, even in the presence of singularities?

Key findings

• The existence of a rotational symmetric CMC surface spanning two concentric circles in parallel planes is equivalent to the condition $ |a - b| < R - r $, matching the result of Bartnik and Simon for the Dirichlet problem.
• Rotational symmetric CMC surfaces are the only compact spacelike CMC surfaces in L³ bounded by two concentric circles in parallel planes.
• Uniqueness holds for CMC surfaces solving the exterior Dirichlet problem with constant boundary data: such solutions must be rotational symmetric.
• Uniqueness also holds for CMC surfaces with an isolated conical-type singularity: such surfaces must be rotational symmetric.
• The flux of the boundary curve $ \Gamma(r,a) $ is preserved between two CMC surfaces with same $ H $, and equality of flux and tangency imply global coincidence.
• The tangency principle, combined with flux and maximum principle arguments, proves that any two CMC surfaces with same $ H $, same boundary flux, and contact at infinity or boundary must be identical.