Combinatorial Yamabe flow on surfaces

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Abstract

In this paper we develop an approach to conformal geometry of piecewise flat metrics on manifolds. In particular, we formulate the combinatorial Yamabe problem for piecewise flat metrics. In the case of surfaces, we define the combinatorial Yamabe flow on the space of all piecewise flat metrics associated to a triangulated surface. We show that the flow either develops removable singularities or converges exponentially fast to a constant combinatorial curvature metric. If the singularity develops, we show that the singularity is always removable by a surgery procedure on the triangulation. We conjecture that after finitely many such surgery changes on the triangulation, the flow converges to the constant combinatorial curvature metric as time approaches infinity.

Original languageEnglish (US)
Pages (from-to)765-780
Number of pages16
JournalCommunications in Contemporary Mathematics
Volume6
Issue number5
DOIs
StatePublished - Oct 1 2004

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Keywords

  • Curvature flow
  • PL metrics
  • PL scalar curvature
  • Surfaces

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