A maximum principle for combinatorial Yamabe flow

By David Glickenstein. 2002

57 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ef4c989dcffde9b76fa6150584ddd5ad1bb65e8d

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In Applied Mathematics

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

A combinatorial Yamabe flow in three dimensions (2005, background): “If we define the curvature of a surface to be ki = 2π− ∑ γijk then the formula for evolution of the face angles implies that the evolution of curvature is in fact parabolic in the usual sense of Laplacians on graphs (this is studied in greater detail in [10]).”
Discrete quasi-Einstein metrics and combinatorial curvature flows in 3-dimension (2013, background): “For any conformal tetrahedron {i, j, k, l} ∈ T , it had been shown in the appendix of [7] (or see [4] and [14]) that the following 4× 4 matrix”
Geometric triangulations and discrete Laplacians on manifolds (2005, background): “It also appears in [20] [21] in three dimensions, where Thurston triangulations are considered such that dij depend only on i.”
COMBINATORIAL YAMABE FLOW ON SURFACES (2003, methodology): “The work [8] follows the approach of [5] and defines a combinatorial Yamabe flow for ball packing metrics.”
Discrete conformal variations and scalar curvature on piecewise flat two and three dimensional manifolds (2009, methodology): “Remark 30 Special cases of this theorem appear in [23][17][18][19] and less refined versions (where only signs and not explicit values are computed for the derivatives) appear in many other places, including [44][39][29][43].”