### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 809-825 |

Number of pages | 17 |

Journal | Topology |

Volume | 44 |

Issue number | 4 |

DOIs | |

State | Published - Jul 1 2005 |

### Keywords

- Curvature flow
- Discrete Riemannian geometry
- Laplacians on graphs
- Maximum principle
- Sphere packing
- Yamabe flow

### ASJC Scopus subject areas

- Geometry and Topology

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## Cite this

*Topology*,

*44*(4), 809-825. https://doi.org/10.1016/j.top.2005.02.002