相关论文: Bounding sectional curvature along a K\"ahler-Ricc…

The limiting behavior of the normalized K\"ahler-Ricci flow for manifolds with positive first Chern class is examined under certain stability conditions. First, it is shown that if the Mabuchi K-energy is bounded from below, then the scalar…

We show that the K\"ahler-Ricci flow on a manifold with positive first Chern class converges to a K\"ahler-Einstein metric assuming positive bisectional curvature and certain stability conditions.

Let $g(t)$, $t\in [0, +\infty)$, be a solution of the normalized K\"ahler-Ricci flow on a compact K\"ahler $n$-manifold $M$ with $c_{1}(M)>0$ and initial metric $g (0)\in 2\pi c_{1}(M)$. If there is a constant $C$ independent of $t$ such…

We show that the scalar curvature is uniformly bounded for the normalized Kahler-Ricci flow on a Kahler manifold with semi-ample canonical bundle. In particular, the normalized Kahler-Ricci flow has long time existence if and only if the…

In this paper, we announce the following results: Let M be a Kaehler-Einstein manifold with positive scalar curvature. If the initial metric has nonnegative bisectional curvature and positive at least at one point, then the K\"ahler-Ricci…

We study the behaviour of the normalized K\"ahler-Ricci flow on complete K\"ahler manifolds of negative holomorphic sectional curvature. We show that the flow exists for all time and converges to a K\"ahler-Einstein metric of negative…

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…

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…

In this paper we prove that for a given K\"ahler-Ricci flow with uniformly bounded Ricci curvatures in an arbitrary dimension, for every sequence of times $t_i$ converging to infinity, there exists a subsequence such that $(M,g(t_i + t))\to…

We show that for any solution to the K\"ahler-Ricci flow with positive bisectional curvature on a compact K\"ahler manifold $M^n$, the bisectional curvature has a uniform positive lower bound. As a consequence, the solution converges…

Let $g(t)$ with $t\in [0,T)$ be a complete solution to the Kaehler-Ricci flow: $\frac{d}{dt}g_{i\bar j}=-R_{i\bar j}$ where $T$ may be $\infty$. In this article, we show that the curvatures of $g(t)$ is uniformly bounded if the solution…

We study the Kahler-Ricci flow on Fano manifolds. We show that if the curvature is bounded along the flow and if the manifold is K-polystable and asymptotically Chow semistable, then the flow converges exponentially fast to a…

We prove the convergence of K\"ahler-Ricci flow with some small initial curvature conditions. As applications, we discuss the convergence of K\"ahler-Ricci flow when the complex structure varies on a K\"ahler-Einstein manifold.

We study the uniqueness problem for the K\"ahler-Ricci flow with a conical initial condition. Given a complete gradient expanding K\"ahler-Ricci soliton on a non compact manifold with quadratic curvature decay, including its derivatives, we…

In this note, we prove that on an $n$-dimensional compact toric manifold with positive first Chern class, the K\"ahler-Ricci flow with any initial $(S^1)^n$-invariant K\"ahler metric converges to a K\"ahler-Ricci soliton. In particular, we…

In this paper, we prove that the $L^4$-norm of Ricci curvature is uniformly bounded along a K\"ahler-Ricci flow on any minimal algebraic manifold. As an application, we show that on any minimal algebraic manifold $M$ of general type and…

In this paper, we establish several sufficient and necessary conditions for the convergence of a K\"ahler-Ricci flow, on a K\"ahler manifold with positive first Chern class, to a K\"ahler-Einstein metric (or a shrinking K\"ahler-Ricci…

In this paper, we show that starting from a geodesic ball $\overline{B_{r_0}}(0)$ in $\mathbb{H}^n$, for $n\geq3$, with prescribed non-decreasing rotationally symmetric mean curvature and the fixed conformal class $[g_{\mathbb{S}^{n-1}}]$…

B Wilking has recently shown that one can associate a Ricci flow invariant cone of curvature operators $C(S)$, which are nonnegative in a suitable sense, to every $Ad_{SO(n,\C)}$ invariant subset $S \subset {\bf so}(n,\C)$. For curvature…

Consider the unnormalized Ricci flow $(g_{ij})_t = -2R_{ij}$ for $t\in [0,T)$, where $T < \infty$. Richard Hamilton showed that if the curvature operator is uniformly bounded under the flow for all times $t\in [0,T)$ then the solution can…