Related papers: Long-time behavior of awesome homogeneous Ricci fl…

This paper investigates the short-time existence and uniqueness of Ricci flow solutions on Finsler manifolds. The main results of this paper are theorems demonstrating the short-time existence of the flow solution for $n$-dimensional…

This paper investigates the question of stability for a class of Ricci flows which start at possibly non-smooth metric spaces. We show that if the initial metric space is Reifenberg and locally bi-Lipschitz to Euclidean space, then two…

In this work, we use the Ricci flow approach to study the gap phenomenon of Riemannian manifolds with non-negative curvature and sub-critical scaling invariant curvature decay. The first main result is a quantitative Ricci flow existence…

We show that every finite volume hyperbolic manifold of dimension greater or equal to 3 is stable under rescaled Ricci flow, i.e. that every small perturbation of the hyperbolic metric flows back to the hyperbolic metric again. Note that we…

In this work, we study metrics which are both homogeneous and Ricci soliton. If there exists a transitive solvable group of isometries on a Ricci soliton, we show that it is isometric to a solvsoliton. Moreover, unless the manifold is flat,…

We introduce singular Ricci flows, which are Ricci flow spacetimes subject to certain asymptotic conditions. We consider the behavior of Ricci flow with surgery starting from a fixed initial compact Riemannian 3-manifold, as the surgery…

In this article, we study complete Type I ancient Ricci flows with positive sectional curvature. Our main results are as follows: in the complete and noncompact case, all such ancient solutions must be noncollapsed on all scales; in the…

We review recent results relating linear stability to dynamical stability and the scalar curvature rigidity of Einstein manifolds. We discuss closed and open Einstein manifolds as well as complete noncompact Einstein manifolds which are…

The geometric flow theory and its applications turned into one of the most intensively developing branches of modern geometry. Here, a brief introduction to Finslerian Ricci flow and their self-similar solutions known as Ricci solitons are…

In this paper we establish the short-time existence and uniqueness theorem for hyperbolic geometric flow, and prove the nonlinear stability of hyperbolic geometric flow defined on the Euclidean space with dimension larger than 4. Wave…

In this paper, we study the Bakry-\'Emery Ricci flow on finite graphs. Our main result is the local existence and uniqueness of solutions to the Ricci flow. We prove the long-time convergence or finite-time blow up for the Bakry-\'Emery…

In this work, we discuss the stability of the pluriclosed flow and generalized Ricci flow. We proved that if the second variation of generalized Einstein--Hilbert functional is nonpositive and the infinitesimal deformations are integrable,…

We show the noninheritance of the completeness of the noncompact Yamabe flow. Our main theorem states the existence of a long time solution which is complete for each time and converges to an incomplete Riemannian metric. This shows the…

In this paper, we study a combinatorial Ricci flow on closed pseudo $3$-manifolds $(M,\mathcal{T})$. We prove that if every edge in the triangulation $\mathcal{T}$ has valence at least $9$, then the combinatorial Ricci flow converges…

In this paper, we study the singularities of two extended Ricci flow systems --- connection Ricci flow and Ricci harmonic flow using newly-defined curvature quantities. Specifically, we give the definition of three types of singularities…

In this paper, we consider Ricci flows admitting closed and smooth tangent flows in the sense of Bamler [Bam20c]. The tangent flow in question can be either a tangent flow at infinity for an ancient Ricci flow, or a tangent flow at a…

In the third part of this series of papers, we establish several topological results that will become important for studying the long-time behavior of Ricci flows with surgery. In the first part of this paper we recall some elementary…

This paper studies the normalized Ricci flow on surfaces with conical singularities. It's proved that the normalized Ricci flow has a solution for a short time for initial metrics with conical singularities. Moreover, the solution makes…

We derive a logarithmic Sobolev inequality along the Ricci flow without any restriction on time, which depends only on the initial metric via rudimentary geometric data, assuming only that a certain first eigenvalue is positive. As a…

In this paper we consider $4$-dimensional steady soliton singularity models, i.e., complete steady gradient Ricci solitons that arise as the rescaled limit of a finite time singular solution of the Ricci flow on a closed $4$-manifold. In…