arXiv research

Enhances understanding of Kähler-Ricci flow singularities.

In this paper, we study the long-term behavior of the conical Kähler-Ricci flow on Fano manifold $M$. First, based on our work of locally uniform regularity for the twisted Kähler-Ricci flows, we obtain a long-time solution to the conical Kähler-Ricci flow by limiting a sequence of these twisted flows. Second, we study…

Study K-R flow on Hirzebruch surfaces, showing tangent flows are K-R flows with orbifold singularities.

Proof of flow convergence on Fano manifolds.

Survey on Kähler-Ricci flow solutions.

We introduce a flow of Kähler structures over Fano manifolds with formal limit at infinite time a Kähler-Ricci soliton. This flow correspond to a Perelman's modified backward Kähler-Ricci type flow that we call Soliton-Kähler-Ricci flow. It can be generated by the Soliton-Ricci flow. We assume that the Soliton-Ricci fl…

We introduce the conical Kähler-Ricci flow modified by a holomorphic vector field. We construct a long-time solution of the modified conical Kähler-Ricci flow as the limit of a sequence of smooth Kähler-Ricci flows.

Clarifies definition of polarized canonical radius in Kahler Ricci flow.

Ancient solutions to Kähler Ricci flow classified completely.

Kahler-Ricci flow long-time behavior and initial data

Kähler-Ricci flow shows type II singularity on Fano threefolds.

Kähler-Ricci flow on Kähler manifolds converges to negative Kodaira dimension

In this paper, we prove the existence of a Kahler Ricci soliton on any smooth Fano horospherical manifold by a study of the Kahler-Ricci flow. Indeed, we prove that the renormalized Kahler Ricci flow converges in the sense of Cheeger Gromov and that this limit is a Kahler-Ricci soliton.

The paper analyzes Kähler-Ricci flow near Kähler-Ricci solitons under complex structure deformation.

Study shows flow convergence to smooth K-Ricci outside a divisor with cusp singularity.

Paper shows regularizing flow for conical Kähler-Ricci equations.

In this note, a modified Kähler-Ricci flow is introduced and studied. The main point is to show the flexibility of Kähler-Ricci flow and summarize some useful techniques.

Proves estimates for Kähler-Ricci flow solutions.

The Kähler-Ricci flow's singularities are analyzed with bounds and convergence results.

Paper studies Kähler-Ricci flow convergence on Fano manifolds.

Paper studies limits of Kähler-Ricci flow on Fano G-manifolds.

Quantizes Kähler-Ricci flow for Fano manifolds.

The Kähler-Ricci flow near conical singularities is described with a $C/t$ curvature bound.

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

Kähler-Ricci flow on spherical Fano manifolds converges to a soliton.

Weak base-point freeness leads to Kähler-Ricci flow diameter bounds.

Uniform Laplace comparison for Kähler Ricci flow on Fano manifolds.

This paper proves uniqueness of Kähler-Ricci flow on non-compact manifolds.

In this short note we announce a regularity theorem for Kähler-Ricci flow on a compact Fano manifold (Kähler manifold with positive first Chern class) and its application to the limiting behavior of Kähler-Ricci flow on Fano 3-manifolds. Moreover, we also present a partial $C^0$ estimate to the Kähler-Ricci flow under …

Kähler-Ricci flow smooths out positive closed currents with divisorial singularities

In this note, we prove that on an $n$-dimensional compact toric manifold with positive first Chern class, the Kähler-Ricci flow with any initial $(S^1)^n$-invariant Kähler metric converges to a Kähler-Ricci soliton. In particular, we give another proof for the existence of Kähler-Ricci solitons on a compact toric manif…

Study Kähler-Ricci flow on manifolds with singularities.

We investigate the limiting behavior of the unnormalized Kahler-Ricci flow on a Kahler manifold with a polarized initial Kahler metric. We prove that the Kahler-Ricci flow becomes extinct in finite time if and only if the manifold has positive first Chern class and the initial Kahler class is proportional to the first …

We investigate the Kähler-Ricci flow modified by a holomorphic vector field. We find equivalent analytic criteria for the convergence of the flow to a Kähler-Ricci soliton. In addition, we relate the asymptotic behavior of the scalar curvature along the flow to the lower boundedness of the modified Mabuchi energy.

In this paper, we prove that Kähler-Ricci flow converges to a Kähler-Einstein metric (or a Kähler-Ricci soliton) in the sense of Cheeger-Gromov as long as an initial Kähler metric is very closed to $g_{KE}$ (or $g_{KS}$) if a compact Kähler manifold with $c_1(M)>0$ admits a Kähler Einstein metric $g_{KE}$ (or a Kähler-…

We prove the existence and uniqueness of the weak Kahler-Ricci flow on projective varieties with log terminal singularities. It is also shown that the weak Kahler-Ricci flow can be uniquely continued through divisorial contractions and flips if they exist. We then propose an analytic version of the Minimal Model Progra…

The paper studies how certain currents can induce metric structures from Kähler-Ricci flows.

In this paper, the author has considered the hyperbolic Kahler-Ricci flow introduced by Kong and Liu [11], that is, the hyperbolic version of the famous Kahler-Ricci flow. The author has explained the derivation of the equation and calculated the evolutions of various quantities associated to the equation including the…

Kähler-Ricci flow singularity type is independent of initial metric.

The Kähler-Ricci flow smooths positive currents on Kähler manifolds.

In this note, we show that the conical Kähler-Ricci flows introduced in \cite{CYW} exist for all time $t\in [0,\infty)$ in the weak sense. As a key ingredient of the proof, we show that a conical Kähler-Ricci flow is actually the limit of a sequence of smooth Kähler-Ricci flows.

Study shows continuity and geometric regularity of Kähler-Ricci flow blow-up limits.

Proves Kähler-Ricci shrinkers are complex analytic varieties.

In this paper, by limiting twisted conical Kähler-Ricci flows, we prove the long-time existence and uniqueness of cusp Kähler-Ricci flow on compact Kähler manifold $M$ which carries a smooth hypersurface $D$ such that the twisted canonical bundle $K_M+D$ is ample. Furthermore, we prove that this flow converge to a uniq…

Kähler-Ricci flows' tangent cones are algebraic varieties.

Study of Kähler-Ricci flow on toric Fano varieties with preserved symplectic condition.

The paper studies a degenerate equation related to Kähler-Ricci flow on symplectic quotients.

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 shrinking Kahler-Ricci solitons. Specifically, the singularity model for these so…