Equivalence of Simplicial Ricci Flow and Hamilton's Ricci Flow for 3D Neckpinch Geometries

TL;DR

This paper demonstrates that simplicial Ricci flow equations in 3D converge to Hamilton's continuum Ricci flow equations for neckpinch geometries, confirming their equivalence in the continuum limit.

Contribution

It establishes the convergence of simplicial Ricci flow to continuum Ricci flow in 3D neckpinch geometries, extending previous numerical and analytical results.

Findings

SRF equations reproduce continuum RF solutions including neckpinch singularities

SRF equations converge to continuum RF equations in the continuum limit

The approach confirms the equivalence of SRF and RF for 3D geometries

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 greater than or equal to 2. The structure of the simplicial Ricci flow (SRF) equations are 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.