A Maximum Principle for Combinatorial Yamabe Flow

Details A Maximum Principle for Combinatorial Yamabe Flow A Maximum Principle for Combinatorial Yamabe Flow

2002-11-13

arxiv.org/abs/math/0211195v3

Topology 44 (2005) 809-825

Mathematics

Metric Geometry

20 pages, this is an almost entirely different paper. Some elements of the old version are in the paper arxiv:math.MG/0506182

Scientific paper

This article studies a discrete geometric structure on triangulated manifolds and an associated curvature flow (combinatorial Yamabe flow). The associated evolution of curvature appears to be like a heat equation on graphs, but it can be shown to not satisfy the maximum principle. The notion of a parabolic-like operator is introduced as an operator which satisfies the maximum principle, but may not be parabolic in the usual sense of operators on graphs. A maximum principle is derived for the curvature of combinatorial Yamabe flow under certain assumptions on the triangulation, and hence the heat operator is shown to be parabolic-like. The maximum principle then allows a characterization of the curvature as well was a proof of long term existence of the flow.

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