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 ≥ 2. The structure of the simplicial Ricci flow (SRF) equations is 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. Exploring simplicial Ricci flow in 3DHamilton's Ricci flow (RF) continues to yield new insights into problems in pure and applied mathematics and proves to be a useful tool across a broad spectrum of engineering fields [1,2,3,4]. Here the time evolution of the metric is proportional to the Ricci tensor,Hamilton showed that this yields a forced diffusion equation for the curvature, e.g. the scalar curvature evolves asThe bulk of the applications of this curvature flow have utilized the numerical evolution of piecewise-flat simplicial 2-surfaces [5,6]. It is a widely accepted verity in computational science that a geometry with complex topology is most naturally represented in a coordinate-free way by an unstructured mesh. This is apparent in the engineering applications utilizing arXiv: 1404.4055 333

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. I. EXPLORING SIMPLICIAL RICCI FLOW IN 3DHamilton's Ricci flow (RF) has yielded new insights into pure and applied mathematics as well as engineering fields [1][2][3][4][5]. Here the time evolution of the metric is proportional to the Ricci tensor,and yields a forced diffusion equation for the curvature; i.e., the scalar curvature evolves aṡThe bulk of the applications of this curvature flow technique have been limited to the numerical evolution of piecewiseflat simplicial 2-surfaces [6]. It is well established that, ordinarily, a geometry with complex topology is most naturally represented in a coordinate-free way by unstructured meshes. e.g. finite volume [7], finite element [8], in general relativity by Regge calculus [9,10] and for electrodynamics by discrete exterior calculus [11]. While the utility of piecewise-flat simplicial geometries in analyzing the RF of 2-dimensional geometries is well established and proven to be effective [6,30], one expects a wealth of exciting new applications in 3 and higher dimensions. Here we explore the utility of RF in three and higher dimensions. However, we first note that the applications of discrete RF in two dimensions arise from its diffusive curvature properties and from the uniformization theorem, that every simply connected Riemann surface evolves under RF to one of three constant curvature surfaces -a sphere, a Euclidean plane or a hyperbolic plane. RF on surfaces is perhaps the only general method to engineer a metric for a surface given only its curvature [6]. In three dimensions the uniformization theorem yields to the geometrization conjecture of Thurston suggesting that...

Abstract:We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data across a discrete sample of times. We analyze the topological signals of geometric criticality obtained numerically from the application of persistent homology to models manifesting singularities under Ricci flow. The results we obtain for these numerical models suggest that the topological signals distinguish global singularity formation (collapse to a round point) from local singularity formation (neckpinch). Finally, we discuss the interpretation and implication of these results and future applications.

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