Related papers: Cusp K\"ahler-Ricci flow on compact K\"ahler manif…

We study the K\"ahler-Ricci flow on compact K\"ahler manifolds whose canonical bundle is big. We show that the normalized K\"ahler-Ricci flow has long time existence in the viscosity sense, is continuous in a Zariski open set, and converges…

We prove that, under a semi-ampleness type assumption on the twisted canonical line bundle, the conical K\"ahler-Ricci flow on a minimal elliptic K\"ahler surface converges in the sense of currents to a generalized conical K\"ahler-Einstein…

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…

We show that the normalized K\"ahler-Ricci flow on a compact K\"ahler manifold with semiample canonical bundle converges in the Gromov-Hausdorff topology to the metric completion of the twisted K\"ahler-Einstein metric on the canonical…

Let $\overline{M}$ be a compact complex manifold with smooth K\"ahler metric $\eta$, and let $D$ be a smooth divisor on $\overline{M}$. Let $M=\overline{M}\setminus D$ and let $\hat{\omega}$ be a Carlson-Griffiths type metric on $M$. We…

In this paper, we study the limit behavior of the conical K\"ahler-Ricci flow as its cone angle tends to zero. More precisely, we prove that as the cone angle tends to zero, the conical K\"ahler-Ricci flow converges to a unique…

We establish the existence of K\"ahler-Ricci flow on pseudoconvex domains with general initial metric without curvature bounds. Moreover we prove that this flow is simultaneously complete, and its normalized version converge to the complete…

We show that on smooth minimal surfaces of general type, the K\"ahler-Ricci flow starting at any initial K\"ahler metric converges in the Gromov-Hausdorff sense to a K\"ahler-Einstein orbifold surface. In particular, the diameter of the…

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 existence of K\"ahler-Einstein metrics on a compact K\"ahler manifold has been the subject of intensive study over the last few decades, following Yau's solution to Calabi's conjecture. The Ricci flow, introduced by Richard Hamilton has…

For the K\"ahler-Ricci flow on a compact K\"ahler manifold with semi-ample canonical line bundle, we prove the singularity type at infinity does not depend on the choice of the initial metric. We also provide new simple proofs for some…

We generalize the maximal time existence of K\"{a}hler-Ricci flow in Tian-Zhang and Song-Tian to conical case. Furthermore, if the twisted canonical bundle $K_{M}+(1-\beta)[D]$ is big or big and nef, we can expect more on the limit…

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 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.

Given a compact K\"ahler manifold $X$ and a closed, positive $(1,1)$-current $T$ on $X$, we find sufficient conditions for $T$ to induce a metric structure $(X,d_T)$ which is the Gromov-Hausdorff limit of compact K\"ahler manifolds either…

We study the K\"ahler-Ricci flow on a class of projective bundles $\mathbb{P}(\mathcal{O}_\Sigma \oplus L)$ over compact K\"ahler-Einstein manifold $\Sigma^n$. Assuming the initial K\"ahler metric $\omega_0$ admits a U(1)-invariant momentum…

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 work, we study the H\"older regularity of the K\"ahler- Ricci flow on compact K\"ahler manifolds with semi-ample canonical line bundle. By adapting the method in the work of Hein-Tosatti on collapsing Calabi-Yau metrics, we…

Starting with a model conical K\"ahler metric, we prove a uniform scalar curvature bound for solutions to the conical K\"ahler-Ricci flow assuming a semi-ampleness type condition on the twisted canonical bundle. In the proof, we also…

In this paper, we prove the long-time existence and uniqueness of the conical K\"ahler-Ricci flow with weak initial data which admits $L^{p}$ density for some $p>1$ on Fano manifold. Furthermore, we study the convergence behavior of this…