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

Alsing, Paul M.; Miller, Warner A.; Corne, Matthew; Gu, Xianfeng; Lloyd, Seth; Ray, Shannon L.; Yau, Shing–Tung

doi:10.48550/arxiv.1308.4148

Cited by 2 publications

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“…We proved recently that the dual-edge SRF equations are equivalent to the simplicial-edge SRF equations for this warped-product geometry in Appendix B of [23],…”

Section: Theorem: the Srf Equations For The Frustum Geometry Are The ...mentioning

confidence: 99%

“…This axisymmetric lattice 3-geometry is characterized by edges of two kinds, the axial edges, a i and the cross-sectional sphere edges, s i . In recent work we examined a discrete model where each of the cross-sectional spheres was approximated by an icosahedron, of edge s i [23]. The accuracy of our model was limited by the relatively low spatial resolution of the icosahedron.…”

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confidence: 99%

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Equivalence of simplicial Ricci flow and Hamilton’s Ricci flow for 3D neckpinch geometries

Miller

Alsing

Corne

et al. 2014

Geometry, Imaging and Computing

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Hamilton's Ricci flow (RF) equations were recently expressed in terms of the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry, for d ≥ 2. The structure of the simplicial Ricci flow (SRF) equations is dimensionally agnostic. These SRF equations were tested numerically and analytically in 3D for simple models and reproduced qualitatively the solution of continuum RF equations including a Type-1 neckpinch singularity. Here we examine a continuum limit of the SRF equations for 3D neck pinch geometries with an arbitrary radial profile. We show that the SRF equations converge to the corresponding continuum RF equations as reported by Angenent and Knopf. Exploring simplicial Ricci flow in 3DHamilton's Ricci flow (RF) continues to yield new insights into problems in pure and applied mathematics and proves to be a useful tool across a broad spectrum of engineering fields [1,2,3,4]. Here the time evolution of the metric is proportional to the Ricci tensor,Hamilton showed that this yields a forced diffusion equation for the curvature, e.g. the scalar curvature evolves asThe bulk of the applications of this curvature flow have utilized the numerical evolution of piecewise-flat simplicial 2-surfaces [5,6]. It is a widely accepted verity in computational science that a geometry with complex topology is most naturally represented in a coordinate-free way by an unstructured mesh. This is apparent in the engineering applications utilizing arXiv: 1404.4055 333

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“…We proved recently that the dual-edge SRF equations are equivalent to the simplicial-edge SRF equations for this warped-product geometry in Appendix B of [23],…”

Section: Theorem: the Srf Equations For The Frustum Geometry Are The ...mentioning

confidence: 99%

mentioning

confidence: 99%

Equivalence of simplicial Ricci flow and Hamilton’s Ricci flow for 3D neckpinch geometries

Miller

Alsing

Corne

et al. 2014

Geometry, Imaging and Computing

Self Cite

View full text Add to dashboard Cite

show abstract

“…Miller and his co-workers [8] have used their extensive experience in the Regge calculus to explore numerical simulations of Ricci flow in S 3 . They have also recently reported on neck pinching singularities in 3-dimensions [11]. The works of Kim and Miller both employ simplicial methods to estimate the curvatures.…”

Section: Introductionmentioning

confidence: 99%

Two dimensional axisymmetric smooth lattice Ricci flow

Brewin

2015

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

Cited by 2 publications

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Equivalence of simplicial Ricci flow and Hamilton’s Ricci flow for 3D neckpinch geometries

Equivalence of simplicial Ricci flow and Hamilton’s Ricci flow for 3D neckpinch geometries

Two dimensional axisymmetric smooth lattice Ricci flow

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