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Mathematics > Differential Geometry

arXiv:1308.4148 (math)
[Submitted on 19 Aug 2013 (v1), last revised 28 Apr 2014 (this version, v2)]

Title:Simplicial Ricci Flow: An Example of a Neck Pinch Singularity in 3D

Authors:Paul M. Alsing, Warner A. Miller, Matthew Corne, Xianfeng Gu, Seth Lloyd, Shannon Ray, Shing-Tung Yau
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Abstract:We examine a Type-1 neck pinch singularity in simplicial Ricci flow (SRF) for an axisymmetric piecewise flat 3-dimensional geometry with 3-sphere topology. SRF was recently introduced as an unstructured mesh formulation of Hamilton's Ricci flow (RF). It describes the RF of a piecewise-flat simplicial geometry. In this paper, we apply the SRF equations to a representative double-lobed axisymmetric piecewise flat geometry with mirror symmetry at the neck similar to the geometry studied by Angenent and Knopf (A-K). We choose a specific radial profile and compare the SRF equations with the corresponding finite-difference solution of the continuum A-K RF equations. The piecewise-flat 3-geometries considered here are built of isosceles-triangle-based frustum blocks. The axial symmetry of this model allows us to use frustum blocks instead of tetrahedra. The 2-sphere cross-sectional geometries in our model are regular icosahedra. We demonstrate that, under a suitably-pinched initial geometry, the SRF equations for this relatively low-resolution discrete geometry yield the canonical Type-1 neck pinch singularity found in the corresponding continuum solution. We adaptively remesh during the evolution to keep the circumcentric dual lattice well-centered. Without such remeshing, we cannot evolve the discrete geometry to neck pinch. We conclude with a discussion of future generalizations and tests of this SRF model.
Comments: 15 pages, 7 figures, submitted to Geometry, Imaging and Computation, minor revisions
Subjects: Differential Geometry (math.DG); General Relativity and Quantum Cosmology (gr-qc)
MSC classes: 53C20 (Primary), 58B20 (Secondary)
Cite as: arXiv:1308.4148 [math.DG]
  (or arXiv:1308.4148v2 [math.DG] for this version)
  https://doi.org/10.48550/arXiv.1308.4148
arXiv-issued DOI via DataCite

Submission history

From: Warner A. Miller [view email]
[v1] Mon, 19 Aug 2013 20:00:09 UTC (1,134 KB)
[v2] Mon, 28 Apr 2014 13:30:48 UTC (1,136 KB)
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