Related papers: An $\epsilon$-Regularity Theorem for Non-collapsed…

In this paper, we will establish a regularity theory for the K\"ahler-Ricci flow on Fano $n$-manifolds with Ricci curvature bounded in $L^p$-norm for some $p > n$. Using this regularity theory, we will also solve a long-standing conjecture…

In this short note we announce a regularity theorem for K\"ahler-Ricci flow on a compact Fano manifold (K\"ahler manifold with positive first Chern class) and its application to the limiting behavior of K\"ahler-Ricci flow on Fano…

In this note we prove a new \epsilon-regularity theorem for the Ricci flow. Let (M^n,g(t)) with t\in [-T,0] be a Ricci flow and H_{x} the conjugate heat kernel centered at a point (x,0) in the final time slice. Substituting H_{x} into…

We consider the K\"ahler-Ricci flow $\frac{\partial}{\partial t}g_{i\bar{j}} = g_{i\bar{j}} - R_{i\bar{j}}$ on a compact K\"ahler manifold $M$ with $c_1(M) > 0$, of complex dimension $k$. We prove the $\epsilon$-regularity lemma for the…

We prove the existence and uniqueness of K\"ahler-Einstein metrics on Q-Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is proper. We study analogues of the works of Perelman on…

In this paper, we study the long-term behavior of the conical K\"ahler-Ricci flow on Fano manifold $M$. First, based on our work of locally uniform regularity for the twisted K\"ahler-Ricci flows, we obtain a long-time solution to the…

In this paper we analyze Ricci flows on which the scalar curvature is globally or locally bounded from above by a uniform or time-dependent constant. On such Ricci flows we establish a new time-derivative bound for solutions to the heat…

In this article, we prove an $\epsilon$-regularity theorem for Perelman's reduced volume. We show that on a Ricci flow, if Perelman's reduced volume is close to $1$, then the curvature radius at the base point cannot be too small.

In this paper, we study the uniformly strong convergence of K\"ahler-Ricci flow on a Fano manifold with varied initial metrics and smooth deformation complex structures. As an application, we prove the uniqueness of K\"ahler-Ricci solitons…

The famous Uniformization Theorem states that on closed Riemannian surfaces there always exists a metric of constant curvature for the Levi-Cevita connection. In this article we prove that an analogue of the uniformization theorem also…

We obtain a compactness result for Fano manifolds and K\"ahler Ricci flows. Comparing to the more general Riemannian versions by Anderson and Hamilton, in this Fano case, the curvature assumption is much weaker and is preserved by the…

We study the generalized K\"ahler-Ricci flow with initial data of symplectic type, and show that this condition is preserved. In the case of a Fano background with toric symmetry, we establish global existence of the normalized flow. We…

The Ricci iteration is a discrete analogue of the Ricci flow. According to Perelman, the Ricci flow converges to a Kahler-Einstein metric whenever one exists, and it has been conjectured that the Ricci iteration should behave similarly.…

We study some estimates along the Kahler Ricci flow on Fano manifolds. Using these estimates, we show the convergence of Kahler Ricci flow directly if the $\alpha$-invariant of the canonical class is greater than $\frac{n}{n+1}$. Applying…

In this paper we characterize non-collapsed limits of Ricci flows. We show that such limits are smooth away from a set of codimension $\geq 4$ in the parabolic sense and that the tangent flows at every point are given by gradient shrinking…

In this paper, we study the behavior of Ricci flows on compact orbifolds with finite singularities. We show that Perelman's pseudolocality theorem also holds on orbifold Ricci flow. Using this property, we obtain a weak compactness theorem…

We first prove a uniform integral Laplace comparison result for the K\"ahler Ricci flow on Fano manifolds which depends only on the initial metric. As an application, using Cheeger-Colding theory and previous results by some of the authors,…

We show the properties of the blowup limits of \KRf solutions on Fano surfaces if Riemannian curvature is unbounded. As an application, on every toric Fano surface, we prove that \KRf converges to a K\"ahler Ricci soliton metric if the…

We investigate Riemannian (non-Kahler) Ricci flow solutions that develop finite-time Type-I singularities and present evidence in favor of a conjecture that parabolic rescalings at the singularities converge to singularity models that are…

In this work, we prove uniqueness for complete non-compact Ricci flow with scaling invariant curvature bound. This generalizes the earlier work of Chen-Zhu, Kotschwar and covers most of the example of Ricci flows with unbounded curvature.…