Timelike Constant Mean Curvature Surfaces with Singularities

TL;DR

This paper uses integrable systems to analyze and construct timelike constant mean curvature surfaces in Lorentz-Minkowski space with various generic singularities, expanding the understanding of their boundary behavior.

Contribution

It generalizes the definition of CMC surfaces to include finite, generic singularities and provides a method to construct such surfaces with prescribed singularities.

Findings

Generic singularities include cuspidal edges, swallowtails, and cuspidal cross caps.

Surfaces with prescribed singularities can be constructed by solving a singular geometric Cauchy problem.

The behavior of surfaces at the boundary of the Birkhoff big cell is characterized.

Abstract

We use integrable systems techniques to study the singularities of timelike non-minimal constant mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space. The singularities arise at the boundary of the Birkhoff big cell of the loop group involved. We examine the behaviour of the surfaces at the big cell boundary, generalize the definition of CMC surfaces to include those with finite, generic singularities, and show how to construct surfaces with prescribed singularities by solving a singular geometric Cauchy problem. The solution shows that the generic singularities of the generalized surfaces are cuspidal edges, swallowtails and cuspidal cross caps.