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

Precise Asymptotics of the Ricci Flow Neckpinch

1.1. Antecedents. In virtually all known applications of the Ricci flow, it is valuable to have a good understanding of singularity formation. Heuristically, there are at least three reasons for this. The first is that one expects finite-time singularities to form for a broad spectrum of initial data. Indeed, such singularities are inevitable if the scalar curvature is strictly positive. The se...

Ricci Lower Bound for Kähler–ricci Flow

Applications of Persistent Homology to Simplicial Ricci Flow

Paul M. Alsing∗1, Howard A. Blair†2, Matthew Corne‡1, Gordon Jones§3, Warner A. Miller¶4, Konstantin Mischaikow‖5 and Vidit Nanda∗∗6 Air Force Research Laboratory, Information Directorate, Rome, NY 13441 Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY 13244 Department of Computer Science, University of Michigan, Ann Arbor, MI 48109 Department of Phys...

Topological Signatures of Singularities in Simplicial Ricci Flow

We apply the methods of persistent homology to a selection of two and three– dimensional geometries evolved by simplicial Ricci flow. To implement persistent homology, we construct a triangular mesh for a sample of points. The scalar curvature along the edges of the triangulation, computed as an average of scalar curvatures at the endpoints of the edges, serves as a filtration parameter at each...

The Ricci Flow for Nilmanifolds

We consider the Ricci flow for simply connected nilmanifolds, which translates to a Ricci flow on the space of nilpotent metric Lie algebras. We consider the evolution of the inner product with respect to time and the evolution of structure constants with respect to time, as well as the evolution of these quantities modulo rescaling. We set up systems of O.D.E.’s for some of these flows and des...