Pseudospherical surfaces with singularities

Article

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 analyse 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.

Keywords

Differential geometry Integrable systems Loop groups Pseudospherical surfaces Constant Gauss curvature Singularities 

Mathematics Subject Classification

Primary 53A05 53C43 Secondary 53C42 

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Computer Science, Matematiktorvet, Building 303 BTechnical University of DenmarkKongens LyngbyDenmark

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