We introduce Forman-Ricci curvature and its corresponding flow as characteristics for complex networks attempting to extend the common approach of node-based network analysis by edge-based characteristics. Following a theoretical introduction and mathematical motivation, we apply the proposed network-analytic methods to static and dynamic complex networks and compare the results with establishe...

In this paper, we study the Ricci flow on closed manifolds equipped with warped product metric (N × F, gN + fgF ) with (F, gF ) Ricci flat. Using the framework of monotone formulas, we derive several estimates for the adapted heat conjugate fundamental solution which include an analog of G. Perelman’s differential Harnack inequality in [18].

We provide a proof that nonholonomically constrained Ricci flows of (pseudo) Riemannian metrics positively result into nonsymmetric metrics (as explicit examples, we consider flows of some physically valuable exact solutions in general relativity). There are constructed and analyzed three classes of solutions of Ricci flow evolution equations defining nonholonomic deformations of Taub NUT, Schw...

Walker manifolds of signature (2, 2) have been used by many authors to provide examples of Osserman and of conformal Osserman manifolds of signature (2, 2). We study questions of geodesic completeness and Ricci blowup in this context.

We use numerical techniques to study the formation of singularities in Ricci flow. Comparing the Ricci flows corresponding to a one parameter family of initial geometries on S3 with varying amounts of S2 neck pinching, we find critical behavior at the threshold of singularity formation

In this short article we show that there are no compact three-dimensional Ricci solitons other than spaces of constant curvature. This generalizes a result obtained for surfaces by Hamilton [4]. The proof involves a careful analysis of the ODE for the curvature which is associated to the Ricci flow.

We numerically calculate Perelman’s entropy for a variety of canonical metrics on CP-bundles over products of Fano Kähler-Einstein manifolds. The metrics investigated are Einstein metrics, Kähler-Ricci solitons and quasi-Einstein metrics. The calculation of the entropy allows a rough picture of how the Ricci flow behaves on each of the manifolds in question.

We prove that for a solution (M, g(t)), t ∈ [0, T ), where T < ∞, to the Ricci flow on a complete non-compact Riemannian manifold with the Ricci curvature tensor uniformly bounded by some constant C on M × [0, T ), the curvature tensor stays uniformly bounded on M × [0, T ).

A new Combinatorial Ricci curvature and Laplacian operators for grayscale images are introduced and tested on 2D synthetic, natural and medical images. Analogue formulae for voxels are also obtained. These notions are based upon more general concepts developed by R. Forman. Further applications, in particular a fitting Ricci flow, are discussed.

Let (M, g) be a compact n-dimensional (n 2) manifold with nonnegative Ricci curvature, and if n 3, then we assume that (M, g) × R has nonnegative isotropic curvature. The lower bound of the Ricci flow’s existence time on (M, g) is proved. This provides an alternative proof for the uniform lower bound of a family of closed Ricci flows’ maximal existence times, which was first proved by E. Cabeza...