Xiuxiong Chen

Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities

This is the first of a series of three papers which provide proofs of results announced recently in arXiv:1210.7494.

Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches 2 and completion of the main proof

This is the third and final paper in a series which establish results announced in arXiv:1210.7494. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle approaches 2\pi. We also put all our technical results together to complete the proof of the main theorem that if a K-stable Fano manifold admits a Kahler-Einstein metric.

Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than 2

This is the second of a series of three papers which provide proofs of results announced in arXiv:1210.7494. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle is less than 2\pi. We show that these are in a natrual way projective algebraic varieties. In the case when the limiting variety and the limiting divisor are smooth we show that the limiting metric also has standard cone singularities.

Ricci flow on Kähler-Einstein surfaces

Abstract.In this paper, we prove that if M is a Kähler-Einstein surface with positive scalar curvature, if the initial metric has nonnegative sectional curvature, and the curvature is positive somewhere, then the Kähler-Ricci flow converges to a Kähler-Einstein metric with constant bisectional curvature. In a subsequent paper [7], we prove the same result for general Kähler-Einstein manifolds in all dimension. This gives an affirmative answer to a long standing problem in Kähler Ricci flow: On a compact Kähler-Einstein manifold, does the Kähler-Ricci flow converge to a Kähler-Einstein metric if the initial metric has a positive bisectional curvature? Our main method is to find a set of new functionals which are essentially decreasing under the Kähler Ricci flow while they have uniform lower bounds. This property gives the crucial estimate we need to tackle this problem.

On Conformally Kahler, Einstein Manifolds

We prove that any compact complex surface with positive first Chern class admits an Einstein metric which is conformally related to a Kaehler metric. The key new ingredient is the existence of such a metric on the blow-up of the complex projective plane at two distinct points.

On the Calabi Flow

In this paper, we study the Calabi flow on a polarized Kähler manifold and some related problems. We first give a precise statement on the short time existence of the Calabi flow for any c3,α(M) initial Kähler potential. As an application, we prove a stability result: any metric near a constant scalar curvature Kähler (CscK) metric will flow to a nearby CscK metric exponentially fast. Secondly, we prove that a compactness theorem in the space of the Kähler metrics given uniform Ricci bound and potential bound. As an application, we prove the Calabi flow can be extended once Ricci curvature stays uniformly bounded. Lastly, we prove a removing-singularity result about a weak constant scalar curvature metric in a punctured disc.

A note on uniformization of Riemann surfaces by Ricci flow

We clarify that the Ricci flow can be used to give an independent proof of the uniformization theorem of Riemann surfaces.

Calabi flow in Riemann surfaces revisited: a new point of view

In this paper, we observe a set of functionals of metrics which are all decrease under the Calabi flow and have uniform lower bound along the flow, which give rise to a set of integral estimates on the curvature flow. Using these estimates, together with weak compactness we obtained in previous papers [8] and [10], we prove the long term existence and convergence of the Calabi flow. Thus give a new proof to Chruscials theorem. The set of simple ideas of global integral estimates and concentration compactness should have further implications in other heat flow problems.

Ricci flow on Kähler-Einstein manifolds

This is the continuation of our earlier article [10]. For any Kähler-Einstein surfaces with positive scalar curvature, if the initial metric has positive bisectional curvature, then we have proved (see [10]) that the Kähler-Ricci flow converges exponentially to a unique Kähler-Einstein metric in the end. This partially answers a long-standing problem in Ricci flow: On a compact Kähler-Einstein manifold, does the Kähler-Ricci flow converge to a Kähler-Einstein metric if the initial metric has positive bisectional curvature? In this article we give a complete affirmative answer to this problem.

On the lower bound of energy functionalE 1 (I)—A stability theorem on the Kähler Ricci flow

In the present article, we prove a stability theorem for the Kaehler Ricci flow near the infimum of the functional E1 under the assumption that the initial metric has Ricci >−1 and ⋎Riem÷ bounded. At present stage, our main theorem still need a topological assumption (1.2) which we hope to be removed in subsequent articles. The underlying moral is: If a Kaehler metric is sufficiently closed to a Kaehler Einstein metric, then the Kaehler Ricci flow converges to it. The present work should be viewed as a first step in a more ambitious program of deriving the existence of Kaehler Einstein metrics with an arbitrary energy level, provided that this energy functional has a uniform lower bound in this kaehler class.