Geometry, Imaging and Computing

Volume 1 (2014)

Number 3

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

Pages: 333 – 366

DOI: http://dx.doi.org/10.4310/GIC.2014.v1.n3.a2

Authors

Warner A. Miller (Department of Physics, Florida Atlantic University, Boca Raton, Fl., U.S.A.; and Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

Paul M. Alsing (Air Force Research Laboratory, Information Directorate, Rome, New York, U.S.A.)

Matthew Corne (Air Force Research Laboratory, Information Directorate, Rome, New York, U.S.A.)

Shannon Ray (Department of Physics, Florida Atlantic University, Boca Raton, Fl., U.S.A.)

Abstract

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 $d \geq 2$. 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.

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