Related papers: Dynamical functionals on ancient ARF Ricci flows

We introduce two new functionals on Sasaki manifolds, inspired by the work of Perelman, which are monotonic along the Sasaki-Ricci flow. We relate their gradient flow, via diffeomorphisms preserving the foliated structure of the manifold,…

The Ricci flow equation of a conformally flat Riemannian metric on a closed 2-dimensional configuration space is analysed. It turns out to be equivalent to the classical Hamilton-Jacobi equation for a point particle subject to a potential…

We present two characterizations of smooth compact Ricci flow solutions solely in terms of metrics and measures (one of them only works under positive scalar curvature along the flow); thus, provide weak formulations that are generalized to…

In this paper, we study the evolving behaviors of the first eigenvalue of Laplace-Beltrami operator under the normalized Ricci flow of model geometries. In every Bianchi class, we estimate the derivative of the eigenvalue. Then we construct…

Ricci flow deforms the Riemannian metric proportionally to the curvature, such that the curvature evolves according to a heat diffusion process and eventually becomes constant everywhere. Ricci flow has demonstrated its great potential by…

We consider compact ancient solutions to the three-dimensional Ricci flow which are noncollapsed. We prove that such a solutions is either a family of shrinking round spheres, or it has a unique asymptotic behavior as $t \to -\infty$ which…

We develop different synthetic notions of Ricci flow in the setting of time-dependent metric measure spaces based on ideas from optimal transport. They are formulated in terms of dynamic convexity and local concavity of the entropy along…

We investigate the properties of the combinatorial Ricci flow for surfaces, both forward and backward -- existence, uniqueness and singularities formation. We show that the positive results that exist for the smooth Ricci flow also hold for…

We introduce the notions of `super-Ricci flows' and `Ricci flows' for time-dependent families of metric measure spaces $(X,d_t,m_t)_{t\in I}$. The former property is proven to be stable under suitable space-time versions of mGH-convergence.…

This article reports recent developments of the research on Hamilton's Ricci flow and its applications.

We give a simple proof for the rotational symmetry of ancient solutions of Ricci flow on surfaces. As a consequence we obtain a simple proof of some results of P.Daskalopoulos, R.Hamilton and N.Sesum on the a priori estimates for the…

We consider the normalized Ricci flow evolving from an initial metric which is conformally compactifiable and asymptotically hyperbolic. We show that there is a unique evolving metric which remains in this class, and that the flow exists up…

We show that any manifold admitting a non-collapsed, ancient Ricci flow must have finite fundamental group. This generalizes what was known for $\kappa$-solutions in dimensions 2, 3. We furthermore show that this fundamental group must be a…

We can talk about two kinds of stability of the Ricci flow at Ricci flat metrics. One of them is a linear stability, defined with respect to Perelman's functional $\mathcal{F}$. The other one is a dynamical stability and it refers to a…

In this paper, we study the Ricci flow on a closed manifold and finite time interval $[0,T)~(T < \infty)$ on which certain integral curvature energies are finite. We prove that in dimension four, such flow converges to a smooth Riemannian…

The Ricci flow was introduced by Hamilton and gained its importance through the years. Of special importance is the limiting behavior of the flow and its symmetry properties. Taking this into account, we present a novel normalization for…

In this paper, we show existence and uniqueness of Ricci flow whose initial condition is a compact Alexandrov surface with curvature bounded from below. This requires a weakening of the notion of initial condition which is able to deal with…

We prove that if an ALE Ricci-flat manifold $(M,g)$ is linearly stable and integrable, it is dynamically stable under Ricci flow, i.e. any Ricci flow starting close to g exists for all time and converges modulo diffeomorphism to an ALE…

We show that the set of awesome homogeneous metrics on non-compact manifolds is Ricci flow invariant. Moreover, if the universal cover of such awesome homogeneous space is not contractible the Ricci flow has finite extinction time,…

In this paper, we introduce a new notion of curvature on the edges of a graph that is defined in terms of effective resistances. We call this the Ricci--Foster curvature. We study the Ricci flow resulting from this curvature. We prove the…