Annals of Mathematical Sciences and Applications

Volume 3 (2018)

Number 1

Special issue in honor of Professor David Mumford, dedicated to the memory of Jennifer Mumford

Guest Editors: Stuart Geman, David Gu, Stanley Osher, Chi-Wang Shu, Yang Wang, and Shing-Tung Yau

A realization of Thurston’s geometrization: discrete Ricci flow with surgery

Pages: 31 – 45

DOI: https://dx.doi.org/10.4310/AMSA.2018.v3.n1.a2

Authors

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

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

Shing-Tung Yau (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

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 axiallysymmetric neck pinch geometry by using surgery to explicitly integrate through its Type-1 neck pinch singularity. A cubic-splinebased adaptive mesh was required to complete the evolution. Our 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 Forman’s expression of Ricci curvature on graphs.

Keywords

Ricci flow, discrete ricci flow, discrete geometry, manifold surgery

2010 Mathematics Subject Classification

Primary 53C20, 53C44. Secondary 52C07, 68R10.

Full Text (PDF format)

We wish to thank Rory Conboye and Matthew Corne for stimulating discussions and for their work. We thank Rory Conboye for his help in reformulating the Forman-Ricci flow equations in their current form. PMA would like to acknowledge support of the Air Force Office of Scientific Research. We wish to thank the Information Directorate of the Air Force Research Laboratory and the Griffiss Institute for providing us with an excellent environment for research. This work was supported in part through the VFRP and SFFP program, as well as AFRL grant FA8750-15-2-0047 and AOARD Grant FA2386-17-1-4070. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of AFRL.

Received 12 July 2017

Published 27 March 2018