Simplicial Ricci Flow: An Example of a Neck Pinch Singularity in 3D

TL;DR

This paper demonstrates that simplicial Ricci flow can model a Type-1 neck pinch singularity in 3D axisymmetric geometries, aligning with continuum solutions and highlighting the importance of adaptive remeshing.

Contribution

It introduces a discrete SRF model for 3D neck pinch singularities, validating its accuracy against continuum RF solutions and emphasizing adaptive remeshing.

Findings

SRF reproduces continuum neck pinch singularity

Adaptive remeshing is essential for evolution

Discrete SRF aligns with continuum RF results

Abstract

We examine a Type-1 neck pinch singularity in simplicial Ricci flow (SRF) for an axisymmetric piecewise flat 3-dimensional geometry with 3-sphere topology. 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 2-sphere 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 well-centered. 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.