A Realization of Thurstons Geometrization: Discrete Ricci Flow with Surgery

TL;DR

This paper demonstrates a numerical implementation of Thurston's geometrization for 3D geometries using discrete Ricci flow with surgery, successfully decomposing a pinched geometry into its canonical geometric pieces.

Contribution

It introduces a discrete Ricci flow method with surgery for 3D geometries, enabling explicit realization of Thurston's geometrization in a numerical setting.

Findings

Successfully decomposed a pinched 3D geometry into geometric pieces

Implemented surgery to navigate singularities during flow

Provided insights into curvature structures for graph Ricci curvature

Abstract

Hamilton's Ricci flow (RF) equations were recently expressed in terms of a sparsely-coupled system of autonomous first-order nonlinear differential equations for the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry. More recently, this system of discrete Ricci flow (DRF) equations was further simplified by explicitly constructing the Forman-Ricci tensor associated to each edge, thereby diagonalizing the first-order differential operator and avoiding the need to invert large sparse matrices at each time step. We recently showed analytically and numerically that these equations converge for axisymmetric 3-geometries to the corresponding continuum RF equations. We demonstrate here that these DRF equations yield an explicit numerical realization of Thurston's geometrization procedure for a discrete 3D axially-symmetric neckpinch geometry by using surgery to explicitly integrate through its Type-1 neck pinch singularity. A cubic-spline-based adaptive mesh was required to complete the evolution. Our numerically efficient simulations yield the expected Thurston decomposition of the sufficiently pinched axially symmetric geometry into its unique geometric structure -- a direct product of two lobes, each collapsing toward a 3-sphere geometry. The structure of our curvature may be used to better inform one of the vertex and edge weighting factors that appear in the Forman's expression of Ricci curvature on graphs.