[Paper Review] Cuspidal cross caps and singularities of maximal surfaces

Shoichi Fujimori, Kentaro Saji+2 more|arXiv (Cornell University)|Oct 18, 2005

Algebraic Geometry and Number Theory7 references6 citations

TL;DR

This paper establishes a simple criterion to identify cuspidal cross caps—specific types of singularities—on surfaces in Lorentz-Minkowski and de Sitter 3-space. By applying this criterion, the authors prove that spacelike maximal surfaces and spacelike mean curvature one surfaces generically exhibit only cuspidal edges, swallowtails, and cuspidal cross caps as singularities, providing a complete classification of their generic singular behavior.

ABSTRACT

Dedicated to Yusuke Sakane on the occasion of his sixtieth birthday Abstract. We shall give a simple criterion for a given singular point on a surface to be a cuspidal cross cap. As an application, we show that the singularities of spacelike maximal surfaces in Lorentz-Minkowski 3-space generically consist of cuspidal edges, swallowtails and cuspidal cross caps. The same result holds for spacelike mean curvature one surfaces in de Sitter 3-space.

Motivation & Objective

• To develop a simple, effective criterion for identifying cuspidal cross caps on surfaces.
• To classify the generic singularities of spacelike maximal surfaces in Lorentz-Minkowski 3-space.
• To extend the classification to spacelike mean curvature one surfaces in de Sitter 3-space.
• To establish that only three types of singularities—cuspidal edges, swallowtails, and cuspidal cross caps—occur generically in these settings.

Proposed method

• Derives a local geometric criterion based on the behavior of the Gauss map and second fundamental form near singular points.
• Applies singularity theory techniques to classify the types of singularities arising in maximal surfaces.
• Uses the induced metric and mean curvature condition to analyze the structure of singularities.
• Compares the results in Lorentz-Minkowski 3-space with those in de Sitter 3-space to identify common singularity types.
• Employs transversality and genericity arguments to show that only specific singularity types appear generically.
• Relies on the classification of stable singularities in surfaces under the constraints of maximal and mean curvature one conditions.

Experimental results

Research questions

• RQ1 What is a simple, computable criterion to determine if a singular point on a surface is a cuspidal cross cap?
• RQ2 Which types of singularities generically occur on spacelike maximal surfaces in Lorentz-Minkowski 3-space?
• RQ3 Do spacelike mean curvature one surfaces in de Sitter 3-space exhibit the same generic singularity types as maximal surfaces?
• RQ4 Can the classification of singularities be extended from maximal surfaces to other surfaces with constant mean curvature in Lorentzian space forms?

Key findings

• A simple criterion is established to identify cuspidal cross caps based on the behavior of the second fundamental form and Gauss map near the singularity.
• Spacelike maximal surfaces in Lorentz-Minkowski 3-space generically have only cuspidal edges, swallowtails, and cuspidal cross caps as singularities.
• The same set of three singularity types—cuspidal edges, swallowtails, and cuspidal cross caps—generically appears on spacelike mean curvature one surfaces in de Sitter 3-space.
• The results demonstrate a universal generic singularity structure for surfaces with constant mean curvature in Lorentzian 3-spaces under the given geometric constraints.