Maximal annuli with parallel planar boundaries in the 3-dimensional Lorentz-Minkowski space

TL;DR

This paper classifies maximal annuli with specific boundary conditions in 3D Lorentz-Minkowski space, showing they are either Lorentzian catenoids or Riemann's examples, and explores convexity properties and exceptions.

Contribution

It provides a classification of maximal annuli with parallel planar boundaries in Lorentz-Minkowski space, extending known results and constructing counterexamples.

Findings

Maximal annuli bounded by circles, lines, or cone points are Lorentzian catenoids or Riemann's examples.

The classification extends to annuli with planar ends under the same boundary conditions.

A maximal annulus with a planar end can violate Shiffman's convexity result.

Abstract

We prove that maximal annuli in $\mathbb{L}^{3}$ bounded by circles, straight lines or cone points in a pair of parallel spacelike planes are part of either a Lorentzian catenoid or a Lorentzian Riemann's example. We show that under the same boundary condition, the same conclusion holds even when the maximal annuli have a planar end. Moreover, we extend Shiffman's convexity result to maximal annuli but by using Perron's method we construct a maximal annulus with a planar end where Shiffman type result fails.