Piecewise flat Ricci flow of compact without boundary three-manifolds

TL;DR

This paper demonstrates that a novel piecewise flat numerical method accurately models Ricci flow on various three-manifolds, with results converging to known solutions or flat geometries as mesh resolution improves.

Contribution

The paper introduces a new piecewise flat approach for simulating Ricci flow on three-manifolds and validates its convergence to smooth solutions across different manifold types.

Findings

Convergence to smooth Ricci flow solutions for Nil and Gowdy manifolds.

Flow towards flat metrics for perturbed and embedded three-torus.

Method's effectiveness across various mesh types and resolutions.

Abstract

Using a recently developed piecewise flat method, numerical evolutions of the Ricci flow are computed for a number of manifolds, using a number of different mesh types, and shown to converge to the expected smooth behaviour as the mesh resolution is increased. The manifolds were chosen to have varying degrees of homogeneity, and include Nil and Gowdy manifolds, a three-torus initially embedded in Euclidean four-space, and a perturbation of a flat three-torus. The piecewise flat Ricci flow of the first two are shown to converge to known smooth Ricci flow solutions, with the remaining two flowing asymptotically to flat metrics.