On holomorphic immersions into kähler manifolds of constant holomorphic sectional curvature

We study holomorphic immersions f:X →M from a complex manifoldX into a Kähler manifold of constant holomorphic sectional curvatureM, i.e. a complex hyperbolic space form, a complex Euclidean space form, or the complex projective space equipped with the Fubini-Study metric. ForX compact we show that the tangent sequence splits holomorphically if and only iff is a totally geodesic immersion. ForX not necessarily compact we relate an intrinsic cohomological invariantp(X) onX, viz. the invariant defined by Gunning measuring the obstruction to the existence of holomorphic projective connections, to an extrinsic cohomological invariant(f) measuring the obstruction to the holomorphic splitting of the tangent sequence. The two invariantsp(X) and?(f) are related by a linear map on cohomology groups induced by the second fundamental form. In some cases, especially whenX is a complex surface andM is of complex dimension 4, under the assumption thatX admits a holomorphic projective connection we obtain a sufficient condition for the holomorphic splitting of the tangent sequence in terms of the second fundamental form.     

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.     

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.      

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.     

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...     

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.     

Kähler manifolds with quasi-constant holomorphic curvature

The aim of this article is to classify compact Kähler manifolds with quasi-constant holomorphic sectional curvature.     

K-stability of constant scalar curvature Kähler manifolds

We show that a polarised manifold with a constant scalar curvature Kähler metric and discrete automorphisms is K-stable. This refines the K-semistability proved by S.K. Donaldson.     

On holomorphic isometric immersions of nonhomogeneous Kähler–Einstein manifolds into the infinite dimensional complex projective space

In the paper, we study the existence of holomorphic isometric immersions from nonhomogeneous Kähler–Einstein manifolds into infinite dimensional complex projective space. It can also be regarded as an application of explicit solutions of complex Monge–Ampère equations on some pseudoconvex domains.     

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.     

Kähler parabolicity and the Euler number of compact manifolds of non-positive sectional curvature

Let be a compact Riemannian manifold of non-positive sectional curvature. It is shown that if is homeomorphic to a K?hler manifold, then its Euler number satisfies the inequality . Received March 14, 1998 / Revised August 7, 2000 / Published online December 8, 2000     

Totally geodesic immersions of Kähler manifolds and Kähler Frenet curves

In a given Kähler manifold (M,J) we introduce the notion of Kähler Frenet curves, which is closely related to the complex structure J of M. Using the notion of such curves, we characterize totally geodesic Kähler immersions of M into an ambient Kähler manifold and totally geodesic immersions of M into an ambient real space form of constant sectional curvature .     

Symplectic mean curvature flows in Kähler surfaces with positive holomorphic sectional curvatures

In this paper, we mainly study the mean curvature flow in Kähler surfaces with positive holomorphic sectional curvatures. We prove that if the ratio of the maximum and the minimum of the holomorphic sectional curvatures is less than $2$ , then there exists a positive constant $\delta $ depending on the ratio such that $\cos \alpha \ge \delta $ is preserved along the flow.     

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.     

A uniformization theorem for complete Kähler manifolds with positive holomorphic bisectional curvature

In the theory of complex geometry, one of the famous problems is the following conjecture of Greene and Wu [13] and Yau [33]: Suppose M is a complete noncompact Kähler manifold with positive holomorphic bisectional curvature; then M is biholomorphic to ?ⁿ. In this paper we use the Ricci flow evolution equation to study this conjecture and prove the result that if M has bounded and positive curvature such that the L’ norm of the curvature on geodesic ball is small enough, then the conjecture is true. Our result gives an improvement on the results of Mok et al. [21] and Mok [22].     

On quasi-Sasakian hypersurfaces of Kähler manifolds

We establish several conditions which are necessary for a quasi-Sasakian hypersurface of a Kähler manifold to be minimal.