Homogeneous nearly K?hler manifolds

The structure of nearly K?hler manifolds was studied by Gray in several articles, mainly in Gray (Math Ann 223:233?C248, 1976). More recently, a relevant progress on the subject has been done by Nagy. Among other results, he proved that a complete strict nearly K?hler manifold is locally a Riemannian product of homogeneous nearly K?hler spaces, twistor spaces over quaternionic K?hler manifolds and six-dimensional (6D) nearly K?hler manifolds, where the homogeneous nearly K?hler factors are also 3-symmetric spaces. In the present article, we show some further properties relative to the structure of nearly K?hler manifolds and, using the lists of 3-symmetric spaces given by Wolf and Gray, we display the exhaustive list of irreducible simply connected homogeneous strict nearly K?hler manifolds. For such manifolds, we give details relative to the intrinsic torsion and the Riemannian curvature.     

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.     

Hodge decomposition theorem on strongly K?hler Finsler manifolds

Faran posed an open problem about analysis on complex Finsler spaces: Is there an analogue of the -Laplacian? Is there an analogue of Hodge theory? Under the assumption that (M, F) is a compact strongly K?hler Finsler manifold, we define a -Laplacian on the base manifold. Our result shows that the well-known Hodge decomposition theorem in K?hler manifolds is still true in the more general compact strongly K?hler Finsler manifolds. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Intrinsic capacities on compact Kähler manifolds

We study fine properties of quasiplurisubharmonic functions on compact Kähler manifolds. We define and study several intrinsic capacities which characterize pluripolar sets and show that locally pluripolar sets are globally “quasi-pluripolar.”     

Trees and tensors on Kähler manifolds

We present an organized method to convert between partial derivatives of metrics (functions) and covariant derivatives of curvature tensors (functions) on Kähler manifolds. Basically, it reduces the highly recursive computation in tensor calculus to the enumeration of certain trees with external legs.     

Imaginary Kählerian Killing spinors I

We describe and to some extent characterize a new family of Kähler spin manifolds admitting non-trivial imaginary Kählerian Killing spinors.     

Symplectic Dolbeault operators on Kähler manifolds

For a Kähler manifold $M$ , the “symplectic Dolbeault operators” are defined using the symplectic spinors and associated Dirac operators, in complete analogy to how the usual Dolbeault operators, $\bar{\partial }$ and $\bar{\partial }^*$ , arise from Dirac operators on the canonical complex spinors on $M$ . We give special attention to two special classes of Kähler manifolds: Riemann surfaces and flag manifolds ( $G/T$ for $G$ a simply-connected compact semisimple Lie group and $T$ a maximal torus). For Riemann surfaces, the symplectic Dolbeault operators are elliptic and we compute their indices. In the case of flag manifolds, we will see that the representation theory of $G$ plays a role and that these operators can be used to distinguish (as Kähler manifolds) between the flag manifolds corresponding to the Lie algebras $B_n$ and $C_n$ . We give a thorough analysis of these operators on $\mathbb{C } P^1$ (the intersection of these classes of spaces), where the symplectic Dolbeault operators have an especially interesting structure.     

Kähler magnetic fields on Kähler manifolds of negative curvature

On a Kähler manifold we have natural uniform magnetic fields which are constant multiples of the Kähler form. Trajectories, which are motions of electric charged particles, under these magnetic fields can be considered as generalizations of geodesics. We give an overview on a study of Kähler magnetic fields and show some similarities between trajectories and geodesics on Kähler manifolds of negative curvature.     

Hamiltonian stationary tori in K?hler manifolds

A Hamiltonian stationary Lagrangian submanifold of a K?hler manifold is a Lagrangian submanifold whose volume is stationary under Hamiltonian variations. We find a sufficient condition on the curvature of a K?hler manifold of real dimension four to guarantee the existence of a family of small Hamiltonian stationary Lagrangian tori.     

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).     

Moment maps associated with holomorphic isometric actions on K?hler manifolds

We give a new formula which writes down the moment map for holomorphic isometric action on a K?hler manifold using the motion of a charged particle when the action of is Hamiltonian.     

An Ohsawa-Takegoshi theorem on compact Khler manifolds

We prove two extension theorems of Ohsawa-Takegoshi type on compact Khler manifolds.In our proof,there are many complications arising from the regularization process of quasi-psh functions on compact Khler manifolds,and unfortunately we only obtain particular cases of the expected result.We remark that the two special cases we proved are natural,since they occur in many situations.We hope that the new techniques we develop here will allow us to obtain the general extension result of Ohsawa-Takegoshi type on compact Khler manifolds in a near future.     

The Bergman metric on complete Kähler manifolds

We use the existence of a bounded uniformly Hölder continuous plurisubharmonic exhaustion function to characterize the Bergman completeness of a complete Kähler manifold. As an application, we proved that any simply-connected complete Kähler manifold with sectional curvature bounded above by a negative constant is Bergman complete. Mathematics Subject Classification (2000):32H10Supported by NSFC grant no. 10271089     

Dynamics of automorphisms on compact Kähler manifolds

We study holomorphic automorphisms on compact Kähler manifolds having simple actions on the Hodge cohomology ring. We show for such automorphisms that the main dynamical Green currents admit complex laminar structures (woven currents) and the Green measure is the unique invariant probability measure of maximal entropy.     

Eigenvalue estimates for the Dirac operator on quaternionic K?hler manifolds

We consider the Dirac operator on compact quaternionic K?hler manifolds and prove a lower bound for the spectrum. This estimate is sharp since it is the first eigenvalue of the Dirac operator on the quaternionic projective space. Received April 21, 1998; in final form June 16, 1998     

Local Schr?dinger flow into K?hler manifolds

In this paper we show that there exists a unique local smooth solution for the Cauchy problem of the Schr?dinger flow for maps from a compact Riemannian manifold into a complete K?hler manifold, or from a Euclidean space R^m into a compact K?hler manifold. As a consequence, we prove that Heisenberg spin system is locally well-posed in the appropriate Sobolev spaces.     

Flat nearly Kähler manifolds

We classify flat strict nearly Kähler manifolds with (necessarily) indefinite metric. Any such manifold is locally the product of a flat pseudo-Kähler factor of maximal dimension and a strict flat nearly Kähler manifold of split signature (2m, 2m) with m ≥ 3. Moreover, the geometry of the second factor is encoded in a complex three-form $\zeta \in \Lambda^3 (\mathbb{C}^m)^*We classify flat strict nearly K?hler manifolds with (necessarily) indefinite metric. Any such manifold is locally the product of a flat pseudo-K?hler factor of maximal dimension and a strict flat nearly K?hler manifold of split signature (2m, 2m) with m ≥ 3. Moreover, the geometry of the second factor is encoded in a complex three-form . The first nontrivial example occurs in dimension 4m = 12.