Related papers: Perelman, Poincare, and the Ricci Flow

Thurston's triangulation conjecture asserts that every hyperbolic 3-manifold admits a geometric decomposition into ideal hyperbolic tetrahedra, a result proven only for certain special 3-manifolds. This paper presents combinatorial Ricci…

In this paper, we derive a relative volume comparison estimate along Ricci flow and apply it to studying the Gromov-Hausdorff convergence of K\"ahler-Ricci flow on a minimal manifold. This new estimate generalizes Perelman's no local…

In this paper we have introduced the notion of $*-$ Ricci flow and shown that $*-$ Ricci soliton which was introduced by Kakimakamis and Panagiotid in 2014, is a self similar soliton of the $*-$ Ricci flow. We have also find the deformation…

In the following series of papers we analyze the long-time behavior of 3 dimensional Ricci flows with surgery. Our main result will be that if the surgeries are performed correctly, then only finitely many surgeries occur and after some…

We use Ricci flow to obtain a local bi-Holder correspondence between Ricci limit spaces in three dimensions and smooth manifolds. This is more than a complete resolution of the three-dimensional case of the conjecture of…

We study $n$-dimensional Ricci flows with non-negative Ricci curvature where the curvature is pointwise controlled by the scalar curvature and bounded by $C/t$, starting at metric cones which are Reifenberg outside the tip. We show that any…

In this paper, we establish a framework for the analysis of linear parabolic equations on conical surfaces and use them to study the conical Ricci flow. In particular, we prove the long time existence of the conical Ricci flow for general…

New elementary, self-contained proofs are presented for the topological and the smooth classification theorems of linear flows on finite-dimensional normed spaces. The arguments, and the examples that accompany them, highlight the…

We show that for certain locally collapsing initial data with Ricci curvature bounded below, one could start the Ricci flow for a definite period of time. This provides a Ricci flow smoothing tool, with which we find topological conditions…

In this paper we prove convergence and compactness results for Ricci flows with bounded scalar curvature and entropy. More specifically, we show that Ricci flows with bounded scalar curvature converge smoothly away from a singular set of…

The theme of the article is the application of the Poincare section method for visual classification of attractors in the four-dimensional phase space; the purpose of the study is to introduce consideration of three-dimensional Poincare…

In this lecture notes, we aim at giving an introduction to the K\"ahler-Ricci flow (KRF) on Fano manifolds. It covers some of the developments of the KRF in its first twenty years (1984-2003), especially an essentially self-contained…

In this note we derive an improved no-local-collapsing theorem of Ricci flow under the scalar curvature bound condition along the worldline of the basepoint. It is a refinement of Perelman's no-local-collapsing theorem.

In this paper we prove a compactness result for Ricci flows with bounded scalar curvature and entropy. It states that given any sequence of such Ricci flows, we can pass to a subsequence that converges to a metric space which is smooth away…

We introduce a natural extension of the concept of gradient Ricci soliton: the Ricci almost soliton. We provide existence and rigidity results, we deduce a-priori curvature estimates and isolation phenomena, and we investigate some…

To every Ricci flow on a manifold M over a time interval I, we associate a shrinking Ricci soliton on the space-time M x I. We relate properties of the original Ricci flow to properties of the new higher-dimensional Ricci flow equipped with…

Using quaternions, we give a concise derivation of the Ricci tensor for homogeneous spaces with topology of the 3-dimensional sphere. We derive explicit and numerical solutions for the Ricci flow PDE and discuss their properties. In the…

The Swampland Distance Conjecture postulates the emergence of an infinite tower of massless states when approaching infinite-distance points in moduli space. However, most string backgrounds are supported by fluxes, and therefore depart…

We give biLipschitz models for the Ricci flow on some 4-manifolds (minimal surfaces of general type), exhibiting a combination of expanding and static behavior.

The present paper is devoted to the study a global aspect of the geometry of harmonic mappings and, in particular, infinitesimal harmonic transformations, and represents the application of our results to the theory of Ricci solutions and…