Geometry, Imaging and Computing

Volume 1 (2014)

Number 3

Simplicial Ricci flow: an example of a neck pinch singularity in 3D

Pages: 303 – 331

DOI: http://dx.doi.org/10.4310/GIC.2014.v1.n3.a1

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.; and Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

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

David Gu (Department of Computer Science, Stony Brook University, Stony Brook, New York, U.S.A.)

Seth Lloyd (Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)

Shannon Ray (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

We examine a Type-1 neck pinch singularity in simplicial Ricci flow (SRF) for an axisymmetric piecewise flat 3-dimensional geometry with topology $S^3$. SRF was recently introduced as an unstructured mesh formulation of Hamilton’s Ricci flow (RF). It describes the RF of a piecewise-flat simplicial geometry. In this paper, we apply the SRF equations to a representative double-lobed axisymmetric piecewise flat geometry with mirror symmetry at the neck similar to the geometry studied by Angenent and Knopf (A-K). We choose a specific radial profile and compare the SRF equations with the corresponding finite-difference solution of the continuum A-K RF equations. The piecewise-flat 3-geometries considered here are built of isosceles-triangle-based frustum blocks. The axial symmetry of this model allows us to use frustum blocks instead of tetrahedra. The $S^2$ cross-sectional geometries in our model are regular icosahedra. We demonstrate that, under a suitably-pinched initial geometry, the SRF equations for this relatively low-resolution discrete geometry yield the canonical Type-1 neck pinch singularity found in the corresponding continuum solution. We adaptively remesh during the evolution to keep the circumcentric dual lattice wellcentered. Without such remeshing, we cannot evolve the discrete geometry to neck pinch. We conclude with a discussion of future generalizations and tests of this SRF model.

Full Text (PDF format)