The K¨ahler-Ricci Flow on K¨ahler Manifolds with 2-Non-negative Traceless Bisectional Curvature Operator

The authors show that the 2-non-negative traceless bisectional curvature is preserved along the Kahler-Ricci flow. The positivity of Ricci curvature is also preserved along the Kahler-Ricci flow with 2-non-negative traceless bisectional curvature. As a corol- lary, the Kahler-Ricci flow with 2-non-negative traceless bisectional curvature will converge to a Kahler-Ricci soliton in the sense of Cheeger-Cromov-Hausdorff topology if complex dimension n ≥ 3.     

The K(a)ihler-Ricci Flow on K(a)hler Manifolds with 2-Non-negative Traceless Bisectional Curvature Operator

The authors show that the 2-non-negative traceless bisectional curvature is preserved along the K(a)hler-Ricci flow.The positivity of Ricci curvature is also preserved along the K(a)hler-Ricci flow with 2-non-negative traceless bisectional curvature.As a corollary,the K(a)hler-Ricci flow with 2-non-negative traceless bisectional curvature will converge to a K(a)hler-Ricci soliton in the sense of Cheeger-Gromov-Hausdorff topology if complex dimension n≥3.     

Characterizations of Null Holomorphic Sectional Curvature of G C R-Lightlike Submanifolds of Indefinite Nearly K ¨ahler Manifolds

We obtain the expressions for sectional curvature, holomorphic sectional curvature and holomorphic bisectional curvature of a GCR-lightlike submanifold of an indefinite nearly K ¨ahler manifold and obtain characterization theorems for holo-morphic sectional and holomorphic bisectional curvature. We also establish a condi-tion for a GCR-lightlike submanifold of an indefinite complex space form to be a null holomorphically flat.     

Ricci flow on Kähler manifolds

In this Note, we announce the result that if M is a Kähler–Einstein manifold with positive scalar curvature, if the initial metric has nonnegative bisectional curvature, and the curvature is positive somewhere, then the Kähler–Ricci flow converges to a Kähler–Einstein metric with constant bisectional curvature.     

Maximum principles for real (p, p)-forms on K?hler manifolds

In this paper, we extend the maximum principle for (1, 1)-Hermitian symmetric tensor to a complete K?hler manifold with bounded holomorphic bisectional curvature and nonnegative orthogonal bisectional curvature. We also achieve a maximum principle for real (p, p)-forms on a compact K?hler manifold with nonnegative holomorphic sectional curvature and vanishing Bochner tensor.     

VANISHING THEOREMS FOR ACH KAHLER MANIFOLDS AND HARMONIC MAPS

We discuss a class of complete Kaihler manifolds which are asymptotically complex hyperbolic near infinity. The main result is vanishing theorems for the second L2 cohomology of such manifolds when it has positive spectrum. We also generalize the result to the weighted Poincare inequality case and establish a vanishing theorem provided that the weighted function p is of sub-quadratic growth of the distance function. We also obtain a vanishing theorem of harmonic maps on manifolds which satisfies the weighted Poincare inequality.     

A no expanding breather theorem for noncompact Ricci flows

Annals of Global Analysis and Geometry - In this note we show that, under certain curvature positivity conditions (the weak $${\text {PIC}}-2$$ condition or the nonnegative bisectional curvature...     

On a kähler version of cheeger-gromoll-perelman's soul theorem

In this note, we will prove a Kähler version of Cheeger-Gromoll-Perelman's soul theorem, only assuming the sectional curvature is nonnegative and bisectional curvature is positive at one point.     

An extension of Mok's theorem on the generalized Frankel conjecture

In this paper,we will give an extension of Mok's theorem on the generalized Frankel conjecture under the condition of the orthogonal holomorphic bisectional curvature.     

复射影空间CP^n＋p的Kaehler子流形

本文对复射影空间ＣＰｎ＋ｐ中具有非负全实双截面曲率的ｎ维（ｐ＜ｎ）紧致Ｋａｅｈｌｅｒ子流形作了完全分类．     

Immortal solution of the Ricci flow

For any complete noncompact Kähler manifold with nonnegative and bounded holomorphic bisectional curvature, we provide the necessary and sufficient condition for the immortal solution to the Ricci flow.     

Immortal solution of the Ricci flow

For any complete noncompact Kahler manifold with nonnegative and bounded holomorphic bisectional curvature, we provide the necessary and sufficient condition for the immortal solution to the Ricci flow.     

Ricci flow on Kähler-Einstein surfaces

In this paper, we prove that if M is a K?hler-Einstein surface with positive scalar curvature, if the initial metric has nonnegative sectional curvature, and the curvature is positive somewhere, then the K?hler-Ricci flow converges to a K?hler-Einstein metric with constant bisectional curvature. In a subsequent paper [7], we prove the same result for general K?hler-Einstein manifolds in all dimension. This gives an affirmative answer to a long standing problem in K?hler Ricci flow: On a compact K?hler-Einstein manifold, does the K?hler-Ricci flow converge to a K?hler-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 K?hler Ricci flow while they have uniform lower bounds. This property gives the crucial estimate we need to tackle this problem. Oblatum 8-IX-2000 & 30-VII-2001?Published online: 19 November 2001     

VOLUME GROWTH ESTIMATES OF MANIFOLDS WITH NONNEGATIVE CURVATURE OUTSIDE A COMPACT SET

In this article, using the properties of Busemann functions, the authors prove that the order of volume growth of Kahler manifolds with certain nonnegative holomorphic bisectional curvature and sectional curvature is at least half of the real dimension. The authors also give a brief proof of a generalized Yau＇s theorem.     

The Sasaki–Ricci Flow and Compact Sasaki Manifolds of Positive Transverse Holomorphic Bisectional Curvature

We show that Perelman’s ${\mathcal{W}}$ functional on Kähler manifolds has a natural counterpart on Sasaki manifolds. We prove, using this functional, that Perelman’s results on Kähler–Ricci flow (the first Chern class is positive) can be generalized to Sasaki–Ricci flow, including the uniform bound on the diameter and the scalar curvature along the flow. We also show that positivity of transverse bisectional curvature is preserved along Sasaki–Ricci flow, using Bando and Mok’s methods and results in Kähler–Ricci flow. In particular, we show that the Sasaki–Ricci flow converges to a Sasaki–Ricci soliton when the initial metric has nonnegative transverse bisectional curvature.     

A Remark on Steinness

In this paper, the authors prove that if Mn is a complete noncompact Kahler manifold with a pole p, and its holomorphic bisectional curvature is asymptotically non-negative to p, then it is a Stein manifold.     

一类紧致K¨ahler流形上的Harnack估计

给出了一些紧致~K\"{a}hler~流形上具有和时间相关的位势热方程的正解的Hanack估计.作为应用, 得到了两个~K\"{a}hler-Ricci~流下具有非负双截面曲率的单调熵.     

The Kähler–Ricci flow with positive bisectional curvature

We show that the Kähler–Ricci flow on a manifold with positive first Chern class converges to a Kähler–Einstein metric assuming positive bisectional curvature and certain stability conditions.     

Real hypersurfaces with constant totally real bisectional curvature in complex space forms

In this paper we classify real hypersurfaces with constant totally real bisectional curvature in a non flat complex space form M _m (c), c ≠ 0 as those which have constant holomorphic sectional curvature given in [6] and [13] or constant totally real sectional curvature given in [11].     

On the volume growth of Kählerian manifolds with positive curvature near infinity

In this paper, the authors proved that the order of volume growth of Kählerian manifolds with positive bisectional curvature near infinity is at least half of the real dimension (i.e., the complex dimension).