Timelike Constant Mean Curvature Surfaces with Singularities

David Brander, Martin Svensson

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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.
Original languageEnglish
JournalJournal of Geometric Analysis
Volume24
Issue number3
Pages (from-to)1641-1672
ISSN1050-6926
DOIs
Publication statusPublished - 2014

Keywords

  • Differential geometry
  • Integrable systems
  • Timelike CMC surfaces
  • Singularities
  • Constant mean curvature

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