Equivalence of simplicial Ricci flow and Hamilton’s Ricci flow for 3D neckpinch geometries

Abstract

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

Piecewise flat curvature and Ricci flow in three dimensions

“…These then match the sectional curvatures orthogonal to a M and b M in the smooth manifold. The prism-based piecewise flat decomposition was used effectively in [13] to give the Ricci flow of a 3-cylinder, and was also extended in [16,34] to non-fixed radial lengths along R 1 , producing a dumbbell model, where a good approximation of the Ricci flow behaviour was also found. In order to remove the enforcement of the orthogonality condition however, the prisms must be subdivided into tetrahedra.…”

Topological Signals of Singularities in Ricci Flow

“…We begin with summaries of persistent homology and Ricci flow. Since our treatment of these topics is cursory, we refer the interested reader to [18,19,29,21] for overviews of persistent homology and its applications and to [30,17] for various technical details pertaining to Ricci flow and its discrete analogues.…”

“…Other suitable triangulations of a manifold for numerical evolution of Ricci flow exist (cf. [30]); we have chosen this one for convenience. For each vertex chosen, we obtain the scalar curvature.…”

Piecewise linear manifolds: Einstein metrics and Ricci flows

“…More recently attempts have been made to introduce additional curvature notions. In particular analogues of the Ricci tensor and a Ricci flow [1,10,14,22,25,30,31,32,47,49], a Yamabe flow [18], as well as an analogue of an Einstein space were proposed [10].…”