Almost contact Kähler manifolds of constant holomorphic sectional curvature

We introduce the notion of an almost contact Kähler structure. We also define the holomorphic sectional curvature of the distribution of an almost contact Kähler structure with respect to an interior metric connection and establish relations between the φ-sectional curvature of an almost contact Kähler manifold and the holomorphic sectional curvature of the distribution of an almost contact Kähler structure.     

Schwarz lemma from a Kähler manifold into a complex Finsler manifold

Science China Mathematics - Suppose that M is a complete Kähler manifold such that its holomorphic sectional curvature is bounded from below by a constant and its radial sectional curvature is...     

Pluriclosed Manifolds with Constant Holomorphic Sectional Curvature

Acta Mathematica Sinica, English Series - A long-standing conjecture in complex geometry says that a compact Hermitian manifold with constant holomorphic sectional curvature must be Kähler...     

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.     

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.     

Central Kähler metrics

The determinant of the Ricci endomorphism of a Kähler metric is called its central curvature, a notion well-defined even in the Riemannian context. This work investigates two types of Kähler metrics in which this curvature potential gives rise to a potential for a gradient holomorphic vector field. These metric types generalize the Kähler-Einstein notion as well as that of Bando and Mabuchi (1986). Whenever possible the central curvature is treated in analogy with the scalar curvature, and the metrics are compared with the extremal Kähler metrics of Calabi. An analog of the Futaki invariant is employed, both invariants belonging to a family described in the language of holomorphic equivariant cohomology. It is shown that one of the metric types realizes the minimum of an functional defined on the space of Kähler metrics in a given Kähler class. For metrics of constant central curvature, results are obtained regarding existence, uniqueness and a partial classification in complex dimension two. Consequently, on a manifold of Fano type, such metrics and Kähler-Einstein metrics can only exist concurrently. An existence result for the case of non-constant central curvature is stated, and proved in a sequel to this work.

     

New geometric flows on Riemannian manifolds and applications to Schrödinger-Airy flows

In this paper,a class of new geometric flows on a complete Riemannian manifold is defined. The new flow is related to the generalized(third order) Landau-Lifshitz equation. On the other hand it could be thought of as a special case of the Schr¨odinger-Airy flow when the target manifold is a K¨ahler manifold with constant holomorphic sectional curvature. We show the local existence of the new flow on a complete Riemannian manifold with some assumptions on Ricci tensor. Moreover,if the target manifolds are Einstein or some certain type of locally symmetric spaces,the global results are obtained.     

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.     

On holomorphic sectional curvature of Kähler quotients

Archiv der Mathematik - In this article, we compute the holomorphic sectional curvature of non-singular Kähler quotients. As a corollary, we show that the holomorphic sectional curvature of...     

Almost hermitian manifolds and Osserman condition

Let(M, g, J) be an almost Hermitian manifold. In this paper we study holomorphically nonnegatively(δ,Δ)-pinched almost Hermitian manifolds. In [3] it was shown that for such Kahler manifolds a plane with maximal sectional curvature has to be a holomorphic plane(J-invariant). Here we generalize this result to arbitrary almost Hermitian manifolds with respect to the holomorphic curvature tensorH R and toRK-manifolds of a constant type λ(p). In the proof some estimates of the sectional curvature are established. The results obtained are used to characterize almost Hermitian manifolds of constant holomorphic sectional curvature (with respect to holomorphic and Riemannian curvature tensor) in terms of the eigenvalues of the Jacobi-type operators, i.e. to establish partial cases of the Osserman conjecture. Some examples are studied. The first author is partially supported by SFS, Project #04M03.     

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.     

The Comparison Theorem for Bergman and Kobayashi Metrics on Cartan-Hartogs Domain of the Third Type

In this paper, we give the holomorphic sectional curvature under invariant Kähler metric on a Cartan-Hartogs domain of the third type Y _III (N,q,K) and construct an invariant Kähler metric, which is complete and not less than the Bergman metric, such that its holomorphic sectional curvature is bounded above by a negative constant. Hence we obtain a comparison theorem for the Bergman and Kobayashi metrics on Y _III (N,q,K).     

Properly discontinuous actions on Hilbert manifolds

In this article we study properly discontinuous actions on Hilbert manifolds giving new examples of complete Hilbert manifolds with nonnegative, respectively nonpositive, sectional curvature with infinite fundamental group. We also get examples of complete infinite dimensional Kähler manifolds with positive holomorphic sectional curvature and infinite fundamental group in contrastwith the finite dimensional case and we classify abelian groups acting linearly, isometrically and properly discontinuously on Stiefel manifolds. Finally, we classify homogeneous Hilbert manifolds with constant sectional curvature.     

On Kähler manifolds with harmonic Bochner curvature tensor

We prove that every irreducible Kähler manifold with harmonic Bochner curvature tensor and constant scalar curvature is Kähler–Einstein and that every irreducible compact Kähler manifold with harmonic Bochner curvature tensor and negative semi-definite Ricci tensor is Kähler–Einstein.     

Lie algebras ofH-projective motions of Kähler manifolds of constant holomorphic sectional curvature

In the paper, the Lie algebras of infinitesimalH-projective transformations with2n-dimensional Kähler manifolds of constant holomorphic sectional curvature are considered. It is proved that these algebras are isomorphic to the realification of the complex Lie algebra $sl(n, \mathbb{C})$ , and their local realization in the form of an algebra of vector fields on a manifold is described.     

Hyperkähler Syz Conjecture and Semipositive Line Bundles

Let M be a compact, holomorphic symplectic Kähler manifold, and L a non-trivial line bundle admitting a metric of semipositive curvature. We show that some power of L is effective. This result is related to the hyperkähler SYZ conjecture, which states that such a manifold admits a holomorphic Lagrangian fibration, if L is not big.     

On the Curvature of Kähler Manifolds with Zero Ricci Tensor

Mathematical Notes - The behavior of the modulus of the curvature tensor and of the holomorphic sectional curvature on Ricci-flat Kähler manifolds is investigated.     

Kähler tubes of constant radial holomorphic sectional curvature

We determine (up to holomorphic isometries) the family of Kähler tubes, around totally geodesic complex submanifolds, of constant radial holomorphic sectional curvature when the centreP of the tube is either simply connected or a complex hypersurface withH ¹ (P, R)=0. In the last case, these tubes have the topology of tubular neighbourhoods of the zero section of the complex lines bundles over symplectic manifolds (when they are Kähler) of the Kostant-Souriau prequantization.