Equivalence of Simplicial Ricci Flow and Hamilton's Ricci Flow for 3D Neckpinch Geometries

Miller, Warner A.; Alsing, Paul M.; Corne, Matthew; Ray, Shannon L.

doi:10.48550/arxiv.1404.4055

Cited by 2 publications

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“…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.…”

Section: Preliminaries and Backgroundmentioning

confidence: 99%

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Topological Signals of Singularities in Ricci Flow

Alsing

Blair

Corne

et al. 2017

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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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Section: Preliminaries and Backgroundmentioning

confidence: 99%

“…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.…”

Section: Data Generation and Preparation Algorithmmentioning

confidence: 99%

Topological Signals of Singularities in Ricci Flow

Alsing

Blair

Corne

et al. 2017

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“…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].…”

Section: Introductionmentioning

confidence: 99%

Piecewise linear manifolds: Einstein metrics and Ricci flows

Schrader¹

2016

J. Phys. A: Math. Theor.

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This article provides an attempt to extend concepts from the theory of Riemannian manifolds to piecewise linear spaces. In particular we propose an analogue of the Ricci tensor, which we give the name of an Einstein vector field. On a given set of piecewise linear spaces we define and discuss (normalized) Einstein flows. Piecewise linear Einstein metrics are defined and examples are provided. Criteria for flows to approach Einstein metrics are formulated. Second variations of the total scalar curvature at a specific Einstein space are calculated.Date: April 25, 2018File: PLEEv2.tex.

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Equivalence of Simplicial Ricci Flow and Hamilton's Ricci Flow for 3D Neckpinch Geometries

Cited by 2 publications

References 0 publications

Topological Signals of Singularities in Ricci Flow

Topological Signals of Singularities in Ricci Flow

Piecewise linear manifolds: Einstein metrics and Ricci flows

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