Pseudospherical surfaces with singularities

TL;DR

This paper explores a generalization of pseudospherical surfaces with singularities, classifying them, providing construction methods, and solving a geometric Cauchy problem with uniqueness conditions based on curvature and torsion.

Contribution

It introduces pseudospherical frontals with singularities, classifies their types, and offers methods for constructing and solving the associated singular geometric Cauchy problem.

Findings

Classification of singularities into characteristic and non-characteristic types

Methods for constructing all non-degenerate singularities

Unique solutions for most curves in the Cauchy problem, with exceptions based on curvature and torsion

Abstract

We study a generalization of constant Gauss curvature -1 surfaces in Euclidean 3-space, based on Lorentzian harmonic maps, that we call pseudospherical frontals. We analyze the singularities of these surfaces, dividing them into those of characteristic and non-characteristic type. We give methods for constructing all non-degenerate singularities of both types, as well as many degenerate singularities. We also give a method for solving the singular geometric Cauchy problem: construct a pseudospherical frontal containing a given regular space curve as a non-degenerate singular curve. The solution is unique for most curves, but for some curves there are infinitely many solutions, and this is encoded in the curvature and torsion of the curve.