Related papers: A note on K\"ahler-Ricci soliton

This paper is the first step in a systematic project to study examples of K\"ahler manifolds with positive holomorphic sectional curvature ($H > 0$). Previously Hitchin proved that any compact K\"ahler surface with $H>0$ must be rational…

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 paper, by combining techniques from Ricci flow and algebraic geometry, we prove the following generalization of the classical uniformization theorem of Riemann surfaces. Given a complete noncompact complex two dimensional K\"ahler…

Let $(M, g)$ be a complete non-compact K\"ahler manifold with non-negative and bounded holomorphic bisectional curvature. Extending our techniques developed in \cite{CT3}, we prove that the universal cover $\wt M$ of $M$ is biholomorphic to…

In this paper, we study MRC fibrations of compact K\"ahler manifolds with partially semi-positive curvature. We first prove that a compact K\"ahler manifold is rationally connected if its tangent bundle is BC-$p$ positive for all $1\leq…

In this note, we show that any $n$-dimensional $\kappa$-noncollapsed steady K\"ahler-Ricci soliton with nonnegative bisectional curvature must be flat. The result is an improvement to our former work in \cite{DZ2}.

We investigate four-dimensional gradient shrinking Ricci solitons with positive modified sectional curvature. Our first main result shows that if the norm of the self-dual Weyl tensor and the scalar curvature satisfy a certain sharp…

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 study the moduli spaces of noncollapsed Ricci flow solutions with bounded energy and scalar curvature. We show a weak compactness theorem for such moduli spaces and apply it to study isoperimetric constant control,…

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 consider the K\"ahler-Ricci flow $(X, \omega(t))_{t \in [0,T)}$ on a compact manifold where the time of singularity, $T$, is finite. We assume the existence of a holomorphic map from the K\"ahler manifold $X$ to some analytic variety $Y$…

Compact K\"ahler solvmanifolds are classified up to biholomorphism. A proof of a conjecture Benson and Gordon, that completely solvable compact K\"ahler solvmanifolds are tori is deduced from this. The main ingredient in the proof is a…

We give necessary and sufficient conditions for a K\"ahler equivariant resolution of a K\"ahler cone, with the resolution satisfying one of a number of auxiliary conditions, to admit a unique asymptotically conical (AC) expanding gradient…

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 show that a shrinking Ricci soliton with positive sectional curvature must be compact. This extends a result of Perelman in dimension three and improves a result of Naber in dimension four, respectively.

Let $X$ be a compact K\"ahler manifold. We prove that if $X$ admits a smooth Hermitian metric $\omega$ with quasi-positive second Chern-Ricci curvature $\mathrm{Ric}^{(2)}(\omega)$, then $X$ is projective and rationally connected. In…

In this paper, we establish a "pseudo-effective" version of the holonomy principle for compact K\"{a}hler manifolds with nonnegative holomorphic sectional curvature. As applications, we prove that if a compact complex manifold $M$ admits a…

A result of R. Hamilton asserts that any convex hypersurface in an Euclidian space with pinched second fundamental form must be compact. Partly inspired by this result, twenty years ago, in \cite{Ancient}, Remark 3.1 on page 650, the author…

For homogeneous metrics on the spaces of the title it is shown that the Ricci flow can move a metric of stricly positive sectional curvature to one with some negative sectional curvature and one of positive definite Ricci tensor to one with…

We consider complete K\"ahler manifolds with nonnegative Ricci curvature. The main results are: 1. When the manifold has nonnegative bisectional curvature, we show that $\lim\limits_{r\to\infty}\frac{r^{2}}{vol(B(p, r))}\int_{B(p, r)}S$…