Non-Hermitian Yang-Mills connections

We study Yang-Mills connections on holomorphic bundles over complex K?hler manifolds of arbitrary dimension, in the spirit of Hitchin's and Simpson's study of flat connections. The space of non-Hermitian Yang-Mills (NHYM) connections has dimension twice the space of Hermitian Yang-Mills connections, and is locally isomorphic to the complexification of the space of Hermitian Yang-Mills connections (which is, by Uhlenbeck and Yau, the same as the space of stable bundles). Further, we study the NHYM connections over hyperk?hler manifolds. We construct direct and inverse twistor transform from NHYM bundles on a hyperk?hler manifold to holomorphic bundles over its twistor space. We study the stability and the modular properties of holomorphic bundles over twistor spaces, and prove that work of Li and Yau, giving the notion of stability for bundles over non-K?hler manifolds, can be applied to the twistors. We identify locally the following two spaces: the space of stable holomorphic bundles on a twistor space of a hyperk?hler manifold and the space of rational curves in the twistor space of the ‘Mukai’ dual hyperk?hler manifold.     

Kähler manifolds of quasi-constant holomorphic sectional curvatures

The Kähler manifolds of quasi-constant holomorphic sectional curvatures are introduced as Kähler manifolds with complex distribution of codimension two, whose holomorphic sectional curvature only depends on the corresponding point and the geometric angle, associated with the section. A curvature identity characterizing such manifolds is found. The biconformal group of transformations whose elements transform Kähler metrics into Kähler ones is introduced and biconformal tensor invariants are obtained. This makes it possible to classify the manifolds under consideration locally. The class of locally biconformal flat Kähler metrics is shown to be exactly the class of Kähler metrics whose potential function is only a function of the distance from the origin in ? ⁿ . Finally we show that any rotational even dimensional hypersurface carries locally a natural Kähler structure which is of quasi-constant holomorphic sectional curvatures.     

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.     

乘积复流形上的Szabó度量

设(M_1,α),(M_2,β)均为Hermitian流形,本文证明了积流形M_1×M_2上的复Szabó度量F_ε是Berwald度量,且当α,β为K(?)hler度量时,F_ε是强Kahler-Finsler度量,此外本文还给出了F_ε的全纯曲率的显式表达式．     

乘积复流形上的Szabó度量

设(M_1,α),(M_2,β)均为Hermitian流形,本文证明了积流形M_1×M_2上的复Szabó度量F_ε是Berwald度量,且当α,β为K(?)hler度量时,F_ε是强Kahler-Finsler度量,此外本文还给出了F_ε的全纯曲率的显式表达式．     

The Totally Geodesic Coisotropic Submanifolds in Kähler Manifolds

In this paper, we consider the coisotropic submanifolds in a Kähler manifold of nonnegative holomorphic curvature. We prove an intersection theorem for compact totally geodesic coisotropic submanifolds and discuss some topological obstructions for the existence of such submanifolds. Our results apply to Lagrangian submanifolds and real hypersurfaces since the class of coisotropic submanifolds includes them. As an application, we give a fixed-point theorem for compact Kähler manifolds with positive holomorphic curvature. Also, our results can be further extended to nearly Kähler manifolds.     

A note on the almost-one-half holomorphic pinching

Motivated by a previous work by Zheng and the second-named author, we study pinching constants of compact Kähler manifolds with positive holomorphic sectional curvature. In particular, we prove a gap theorem on Kähler manifolds with almost-one-half pinched holomophic sectional curvature. The proof is motivated by the work of Petersen and Tao on Riemannian manifolds with almost-quarter-pinched sectional curvature.     

Kähler manifolds of quasi-constant holomorphic sectional curvature

In correspondence with the manifolds of quasi-constant sectional curvature defined (cf [5], [9]) in the Riemannian context, we introduce in the K?hlerian framework the geometric notion of quasi-constant holomorphic sectional curvature. Some characterizations and properties are given. We obtain necessary and sufficient conditions for these manifolds to be locally symmetric, Ricci or Bochner flat, K?hler η-Einstein or K?hler-Einstein, etc. The characteristic classes are studied at the end and some examples are provided throughout.      

Kähler-Einstein metrics with positive scalar curvature

In this paper, we prove that the existence of K?hler-Einstein metrics implies the stability of the underlying K?hler manifold in a suitable sense. In particular, this disproves a long-standing conjecture that a compact K?hler manifold admits K?hler-Einstein metrics if it has positive first Chern class and no nontrivial holomorphic vector fields. We will also establish an analytic criterion for the existence of K?hler-Einstein metrics. Our arguments also yield that the analytic criterion is satisfied on stable K?hler manifolds, provided that the partial C ⁰-estimate posed in [T6] is true. Oblatum 12-IV-1996 & 8-XI-1996     

RC-positivity and rigidity of harmonic maps into Riemannian manifolds Dedicated to Professor Lo Yang on the Occasion of His 80th Birthday

In this paper,we show that every harmonic map from a compact K?hler manifold with uniformly RC-positive curvature to a Riemannian manifold with non-positive complex sectional curvature is constant.In particular,there is no non-constant harmonic map from a compact Koahler manifold with positive holomorphic sectional curvature to a Riemannian manifold with non-positive complex sectional curvature.     

On isometries of Kähler manifolds

Summary We show that an isometry between not locally hyperk?hler, locally irreducible K?hler manifolds is either holomorphic or antiholomorphic.     

Some comparison theorems for K?hler manifolds

In this work, we will verify some comparison results on K?hler manifolds. They are: complex Hessian comparison for the distance function from a closed complex submanifold of a K?hler manifold with holomorphic bisectional curvature bounded below by a constant, eigenvalue comparison and volume comparison in terms of scalar curvature. This work is motivated by comparison results of Li and Wang (J Differ Geom 69(1):43–47, 2005).     

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.     

Extremal Kähler metrics on projective bundles over a curve

Let M=P(E) be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle E→Σ over a compact complex curve Σ of genus ?2. Building on ideas of Fujiki (1992) [27], we prove that M admits a Kähler metric of constant scalar curvature if and only if E is polystable. We also address the more general existence problem of extremal Kähler metrics on such bundles and prove that the splitting of E as a direct sum of stable subbundles is necessary and sufficient condition for the existence of extremal Kähler metrics in Kähler classes sufficiently far from the boundary of the Kähler cone. The methods used to prove the above results apply to a wider class of manifolds, called rigid toric bundles over a semisimple base, which are fibrations associated to a principal torus bundle over a product of constant scalar curvature Kähler manifolds with fibres isomorphic to a given toric Kähler variety. We discuss various ramifications of our approach to this class of manifolds.     

Maximal ergodic theorems and applications to Riemannian geometry

We prove new ergodic theorems in the context of infinite ergodic theory, and give some applications to Riemannian and Kähler manifolds without conjugate points. One of the consequences of these ideas is that a complete manifold without conjugate points has nonpositive integral of the infimum of Ricci curvatures, whenever this integral makes sense. We also show that a complete Kähler manifold with nonnegative holomorphic curvature is flat if it has no conjugate points.     

Harmonic two-forms on manifolds with non-negative isotropic curvature

We prove that L ² harmonic two-forms are parallel if a complete manifold (M, g) has the non-negative isotropic curvature. Furthermore, if (M, g) has positive isotropic curvature at some point, then there is no non-trivial L ² harmonic two-form. We obtain that an almost K?hler manifold of non-negative isotropic curvature is K?hler and a symplectic manifold can not admit any almost K?hler structure of positive isotropic curvature.     

Compactness theorems and the Calabi flow on Kähler surfaces with stable tangent bundle

Abstract. In this paper, we prove some compactness theorems and collapse phenomenon on compact K?hler surfaces with stable tangent bundle. We then apply the results to the Calabi flow. More precisely, we prove, under suitable curvature conditions, the longtime existence and asymptotic convergence for solutions of the Calabi flow on compact K?hler surfaces admitting no nonzero holomorphic tangent vector fields and with stable tangent bundle. We also give some examples where the Calabi flow blows up. Received January 7, 1999 / Revised February 2, 2000 / Published online July 20, 2000     

Harmonic and holomorphic 1-forms on compact balanced Hermitian manifolds

On compact balanced Hermitian manifolds we obtain obstructions to the existence of harmonic 1-forms, ∂-harmonic (1,0)-forms and holomorphic (1,0)-forms in terms of the Ricci tensors with respect to the Riemannian curvature and the Hermitian curvature. Necessary and sufficient conditions the (1,0)-part of a harmonic 1-form to be holomorphic and vice versa, a real 1-form with a holomorphic (1,0)-part to be harmonic are found. The vanishing of the first Dolbeault cohomology groups of the twistor space of a compact irreducible hyper-Kähler manifold is shown.     

Harmonic Riemannian maps on locally conformal Kaehler manifolds

We study harmonic Riemannian maps on locally conformal Kaehler manifolds (lcK manifolds). We show that if a Riemannian holomorphic map between lcK manifolds is harmonic, then the Lee vector field of the domain belongs to the kernel of the Riemannian map under a condition. When the domain is Kaehler, we prove that a Riemannian holomorphic map is harmonic if and only if the lcK manifold is Kaehler. Then we find similar results for Riemannian maps between lcK manifolds and Sasakian manifolds. Finally, we check the constancy of some maps between almost complex (or almost contact) manifolds and almost product manifolds.