### 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 2005 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

**A maximum principle for combinatorial Yamabe flow.** / Glickenstein, David A.

Research output: Contribution to journal › Article

*Topology*, vol. 44, no. 4, pp. 809-825. https://doi.org/10.1016/j.top.2005.02.002

}

TY - JOUR

T1 - A maximum principle for combinatorial Yamabe flow

AU - Glickenstein, David A

PY - 2005/7

Y1 - 2005/7

N2 - 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.

AB - 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.

KW - Curvature flow

KW - Discrete Riemannian geometry

KW - Laplacians on graphs

KW - Maximum principle

KW - Sphere packing

KW - Yamabe flow

UR - http://www.scopus.com/inward/record.url?scp=17644370592&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=17644370592&partnerID=8YFLogxK

U2 - 10.1016/j.top.2005.02.002

DO - 10.1016/j.top.2005.02.002

M3 - Article

AN - SCOPUS:17644370592

VL - 44

SP - 809

EP - 825

JO - Topology

JF - Topology

SN - 0040-9383

IS - 4

ER -