arXiv subjects
Publications and source records attributed to Zhenlei Zhang.
We prove that any finite time collapsing Kähler Ricci flow on ruled surfaces develops a Type I singularity, such singularity is modeled on the standard product shrinker $\mathbb{P}^1\times \mathbb{C}$. As an application, we obtain the optimal collapse rate of fibers on ruled surfaces.
We prove the Feldman--Ilmanen--Knopf conjecture for finite time Type I singularities of the Kähler--Ricci flow. More precisely, for any compact Kähler manifold $Y$ and its blow-up $π:\operatorname{Bl}_pY\longrightarrow Y$, if $[ω_0]-Tc_1(M)=π^*[ω_Y]$, then any Type I parabolic blow-up limit of the Kähler Ricci flow along the exceptional divisor is the FIK shrinker $\operatorname{Tot}(\mathcal{O}_{\mathbb{P}^{n-1}}(-1))$.
In this paper, we study compactness, rigidity, and related geometric properties of complete Kähler Ricci shrinkers through the polarized Fano fibration structure.
In this paper, we establish diameter bounds for compact Kähler manifolds equipped with Kähler metrics $ω$, assuming the associated measure lies in a specific Orlicz space and satisfies an integrability condition. Firstly, we prove a priori estimates for solutions of the complex Monge-Ampère equation in Orlicz spaces, encompassing $L^{\infty}$ and stability estimates. This is achieved by employing Kołodziej's approach \cite{Ko98} and the argument of Guo-Phong-Tong-Wang \cite{GuPhToWa21}, respectively. Secondly, building on the work of Guo-Phong-Song-Sturm \cite{GuPhSoSt24-1}, we derive the uniform (local/global) estimates of the Green's function and its gradient for the associated Kähler metric $ω$.
This paper is concerned with a class of the long time Kähler-Ricci flow on a compact Kähler manifold. It is shown that the uniform $μ$-entropy or uniform Sobolev inequality along the normalized Kähler-Ricci flow with semiample canonical bundle. As a consequence, we prove that the scalar curvature of the Kähler metrics along the normalized Kähler-Ricci flow converge to negative Kodaira dimension of the compact Kähler manifold.
We first prove a uniform integral Laplace comparison result for the Kähler 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 give a direct and independent proof of the Hamilton-Tian conjecture on convergence of Kähler-Ricci flows, modulo a codimension 4 singular set. We also expounded on some existing literature on this conjecture.
We introduce and construct a novel type of canonical metric: the semi-flat constant scalar curvature Kähler (semi-flat cscK) current, which naturally arises in Calabi-Yau fibrations. For a given elliptic surface $X$ with a holomorphic section, We explicitly construct the desired semi-flat cscK current and analyze its behavior along singular parts. We establish its uniqueness under the condition that $X$ possesses at least one singular fiber other than of type $I_b$ or $I_b^*$. These results contribute to a geometric uniformization program for elliptic surfaces.
We prove a uniform diameter bound for long time solutions of the normalized Kahler-Ricci flow on an $n$-dimensional projective manifold $X$ with semi-ample canonical bundle under the assumption that the Ricci curvature is uniformly bounded for all time in a fixed domain containing a fibre of $X$ over its canonical model $X_{can}$. This assumption on the Ricci curvature always holds when the Kodaira dimension of $X$ is $n$, $n-1$ or when the general fibre of $X$ over its canonical model is a complex torus. In particular, the normalized Kahler-Ricci flow converges in Gromov-Hausdorff topolopy to its canonical model when $X$ has Kodaira dimension $1$ with $K_X$ being semi-ample and the general fibre of $X$ over its canonical model being a complex torus. We also prove the Gromov-Hausdorff limit of collapsing Ricci-flat Kahler metrics on a holomorphically fibred Calabi-Yau manifold is unique and is homeomorphic to the metric completion of the corresponding twisted Kahler-Einstein metric on the regular part of its base.
We study finite-time collapsing limits of the continuity method. When the continuity method starting from a rational initial Kähler metric on a projective manifold encounters a finite-time volume collapsing, this projective manifold admits a Fano fibration over a lower dimensional base. In this case, we prove the continuity method converges to a singular Kähler metric on the base in the weak sense; moreover, if the base is smooth and the fibration has no singular fibers, then the convergence takes place in Gromov-Hausdorff topology.
In this paper, we derive a relative volume comparison estimate along Ricci flow and apply it to studying the Gromov-Hausdorff convergence of Kähler-Ricci flow on a minimal manifold. This new estimate generalizes Perelman's no local collapsing estimate and can be regarded as an analogue of the Bishop-Gromov volume comparison for Ricci flow.
Let $\{g(t)\}_{t\in [0,T)}$ be the solution of the Ricci flow on a closed Riemannian manifold $M^n$ with $n\geq 3$. Without any assumption, we derive lower volume bounds of the form ${\rm Vol}_{g(t)}\geq C (T-t)^{\frac{n}{2}}$, where $C$ depends only on $n$, $T$ and $g(0)$. In particular, we show that $${\rm Vol}_{g(t)} \geq e^{ Tλ-\frac{n}{2}} \left(\frac{4}{(A(λ-r)+4B)T}\right)^{\frac{n}{2}}\left(T-t\right)^{\frac{n}{2}},$$ where $r:=\inf_{\|ϕ\|_2^2=1} \int_M Rϕ^2 \ d{\rm vol}_{g(0)}$, $λ:=\inf_{\|ϕ\|_2^2=1} \int_M 4|\nablaϕ|^2+Rϕ^2\ d{\rm vol}_{g(0)}$ and $A,B$ are Sobolev constants of $(M,g(0))$. This estimate is sharp in the sense that it is achieved by the unit sphere with scalar curvature $R_{g(0)}=n(n-1)$ and $A=\frac{4}{n(n-2)}ω_n^{-\frac{2}{n}}$, $B=\frac{n-1}{n-2}ω_n^{-\frac{2}{n}}$. On the other hand, if the diameter satisfies ${\rm diam}_{g(t)}\leq c_1\sqrt{T-t}$ and there exist a point $x_0\in M$ such that $R(x_0,t)\leq c_2(T-t)^{-1}$, then we have ${\rm Vol}_{g(t)}\leq C (T-t)^{\frac{n}{2}}$ for all $t>\frac{T}{2}$, where $C$ depends only on $c_1,c_2,n,T$ and $g(0)$.
We obtain a local Sobolev constant estimate for integral Ricci curvature, which enables us to extend several important tools such as the maximal principle, the gradient estimate, the heat kernel estimate and the $L^2$ Hessian estimate to manifolds with integral Ricci lower bounds, without the non-collapsing conditions.
We prove that, on a minimal elliptic Kähler surface of Kodaira dimension one, the continuity method introduced by La Nave and Tian in \cite{LT} starting from any initial Kähler metric converges in Gromov-Hausdorff topology to the metric completion of the generalized Kähler-Einstein metric on its canonical model constructed by Song and Tian in \cite{ST06}.
We present a new and direct proof of the local Neumann isoperimetric inequality on convex domains of a Riemannian manifold with Ricci curvature bounded below.
In this paper, we prove that the $L^4$-norm of Ricci curvature is uniformly bounded along a Kähler-Ricci flow on any minimal algebraic manifold. As an application, we show that on any minimal algebraic manifold $M$ of general type and with dimension $n\le 3$, any solution of the normalized Kähler-Ricci flow converges to the unique singular Kähler-Einstein metric on the canonical model of $M$ in the Cheeger-Gromov topology.
In this paper we investigate the differential geometric and algebro-geometric properties of the noncollapsing limit in the continuity method that was introduced by the first two named authors in \cite{LaTi14}.
In this paper, we will establish a regularity theory for the Kähler-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 for dimension 3. As an application, we give a new proof of the Yau-Tian-Donaldson conjecture for Fano 3-manifolds. The results have been announced in \cite{TiZh12b}.
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 the regularity assumption, which extends previous works on Kähler-Einstein metrics and shrinking Kähler-Ricci solitons (cf. \cite{Ti90}, \cite{DoSu12}, \cite{Ti12}, \cite{PSS12}). The detailed proof will appear in \cite{TiZh13}.