Tight Lagrangian homology spheres in compact homogeneous Kähler manifolds

For any irreducible compact homogeneous Kähler manifold, we classify the compact tight Lagrangian submanifolds which have the ?²-homology of a sphere.     

Hamiltonian stability of Lagrangian tori in toric Kähler manifolds

Let (M,J,ω) be a compact toric Kähler manifold of dim_? M=n and L a regular orbit of the T ⁿ-action on M. In the present paper, we investigate Hamiltonian stability of L, which was introduced by Y.-G. Oh (Invent. Math. 101, 501–519 (1990); Math. Z. 212, 175–192) (1993)). As a result, we prove any regular orbit is Hamiltonian stable when (M,ω)=??ⁿ,ω_FS) and (M,ω)=??ⁿ ₁× ??ⁿ ₂,aω_FS⊕ bω_FS), where ω_FS is the Fubini–Study Kähler form and a and b are positive constants. Moreover, they are locally Hamiltonian volume minimizing Lagrangian submanifolds.     

On Hamiltonian stationary Lagrangian spheres in non-Einstein K?hler surfaces

Hamiltonian stationary Lagrangian spheres in K?hler-Einstein surfaces are minimal. We prove that in the family of non-Einstein K?hler surfaces given by the product Σ₁?×?Σ₂ of two complete orientable Riemannian surfaces of different constant Gauss curvatures, there is only a (non minimal) Hamiltonian stationary Lagrangian sphere. This example, defined when the surfaces Σ₁ and Σ₂ are spheres, is unstable.     

Mean curvature flow of space-like Lagrangian submanifolds in almost para-K?hler manifolds

Given an almost para-K?hler manifold equipped with a metric and para-complex connection, we define a generalized second fundamental form and generalized mean curvature vector of space-like Lagrangian submanifolds. We then show that the deformation induced by this variant of the mean curvature vector field preserves the Lagrangian condition, if the connection satisfies also some Einstein condition. In case the almost para-K?hler structure is integrable, the flow coincides with the classical mean curvature flow in the pseudo-Riemannian context.     

Interpolation in non-positively curved Kähler manifolds

We extend to any complete simply connected Kähler manifold with non-positive sectional curvature some conditions for interpolation in\(\mathbb{C}\) and in the unit disk given by Berndtsson, Ortega-Cerdà and Seip. The main tools are L² estimates and a comparison theorem for the Hessian in Kähler geometry due to Greene, Wu, and Siu, Yau.     

The existence of Hamiltonian stationary Lagrangian tori in K?hler manifolds of any dimension

Hamiltonian stationary Lagrangians are Lagrangian submanifolds that are critical points of the volume functional under Hamiltonian deformations. They are natural generalizations of special Lagrangians or Lagrangian and minimal submanifolds. In this paper, we obtain a local condition that gives the existence of a smooth family of Hamiltonian stationary Lagrangian tori in K?hler manifolds. This criterion involves a weighted sum of holomorphic sectional curvatures. It can be considered as a complex analogue of the scalar curvature when the weighting are the same. The problem is also studied by Butscher and Corvino (Hamiltonian stationary tori in Kahler manifolds, 2008).     

On hyper-Kähler manifolds of typeA ∞

In this paper we shall construct new families of 4m dimensional non-compact complete hyper-Kähler manifolds on whichm dimensional torus acts. In the 4 dimensional case our manifolds should be considered as hyper-Kähler manifolds which correspond to the extended Dynkin diagram of typeA .     

On the structure of co-Kähler manifolds

By the work of Li, a compact co-Kähler manifold $M$ is a mapping torus $K_\varphi $ , where $K$ is a Kähler manifold and $\varphi $ is a Hermitian isometry. We show here that there is always a finite cyclic cover $\overline{M}$ of the form $\overline{M} \cong K \times S^1$ , where $\cong $ is equivariant diffeomorphism with respect to an action of $S^1$ on $M$ and the action of $S^1$ on $K \times S^1$ by translation on the second factor. Furthermore, the covering transformations act diagonally on $S^1, K$ and are translations on the $S^1$ factor. In this way, we see that, up to a finite cover, all compact co-Kähler manifolds arise as the product of a Kähler manifold and a circle.     

Pseudo-holomorphic curves in nearly Kähler manifolds

We study pseudo-holomorphic curves in general nearly Kähler manifolds. For that purpose, we first introduce the fundamental equations of submanifold geometry in terms of the characteristic connection of the nearly Kähler structure. Then we classify pseudo-holomorphic curves with parallel second fundamental form in Chern-flat nearly Kähler manifolds. Moreover, we give a new Simons type identity. As an application of this identity, we show that the closed pseudo-holomorphic curves in Chern-flat nearly Kähler manifolds with a second fundamental form of controlled growth are totally geodesic.     

Immersion of Kähler manifolds in the class of convex submanifolds

It is proved that a real complete convex Kähler submanifold in Euclidean space splits as a metric product of two-dimensional surfaces of positive Gaussian curvature in Euclidean 3-space and a Euclidean subspace. A theorem of V. K. Beloshapka and S. N. Bychkov is generalized to the case of convex submanifolds of any codimension.     

Equivariant Kähler compactifications of homogeneous manifolds

A compact complex manifoldX is an equivariant compactification of a homogeneous manifoldG/H (G a connected complex Lie group,H a closed complex subgroup ofG), if there exists a holomorphic action ofG onX such that theG-orbit of some pointx inX is open and H is the isotropy group ofx. GivenG andH, for some groups (e.g.,G nilpotent) there are necessary and sufficient conditions for the existence of an equivariant Kähler compactification which are proven in this paper.     

Transformations of locally conformally Kähler manifolds

We consider several transformation groups of a locally conformally Kähler manifold and discuss their inter-relations. Among other results, we prove that all conformal vector fields on a compact Vaisman manifold which is neither locally conformally hyperkähler nor a diagonal Hopf manifold are Killing, holomorphic and that all affine vector fields with respect to the minimal Weyl connection of a locally conformally Kähler manifold which is neither Weyl-reducible nor locally conformally hyperkähler are holomorphic and conformal.     

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.     

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.     

Decomposition and minimality of lagrangian submanifolds in nearly Kähler manifolds

We show that Lagrangian submanifolds in six-dimensional nearly Kähler (non-Kähler) manifolds and in twistor spaces Z ⁴ⁿ⁺² over quaternionic Kähler manifolds Q ⁴ⁿ are minimal. Moreover, we prove that any Lagrangian submanifold L in a nearly Kähler manifold M splits into a product of two Lagrangian submanifolds for which one factor is Lagrangian in the strict nearly Kähler part of M and the other factor is Lagrangian in the Kähler part of M. Using this splitting theorem, we then describe Lagrangian submanifolds in nearly Kähler manifolds of dimensions six, eight, and ten.     

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 .     

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.     

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.