The Journal of Geometric Analysis

, Volume 24, Issue 3, pp 1641–1672 | Cite as

Timelike Constant Mean Curvature Surfaces with Singularities

Article

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 behavior 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.

Keywords

Differential geometry Integrable systems Timelike CMC surfaces Singularities Constant mean curvature 

Mathematics Subject Classification (2010)

53A10 53C42 53A35 

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Copyright information

© Mathematica Josephina, Inc. 2013

Authors and Affiliations

  1. 1.Department of Mathematics, MatematiktorvetTechnical University of DenmarkKgs. LyngbyDenmark
  2. 2.Department of Mathematics & Computer Science and CP3-Origins, Centre of Excellence for Particle Physics PhenomenologyUniversity of Southern DenmarkOdense MDenmark

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