Leafwise Brownian Motions and Some Function Theoretic Properties of Laminations

We discuss the value distribution of Borel measurable functions which are subharmonic or meromorphic along leaves on laminations. They are called leafwise subharmonic functions or meromorphic functions respectively. We consider cases that each leaf is a negatively curved Riemannian manifold or Kähler manifold. We first consider the case when leaves are Riemannian with a harmonic measure in L.Garnett sense. We show some Liouville type theorem holds for leafwise subharmonic functions in this case. In the case of laminations whose leaves are Kähler manifolds with some curvature condition we consider the value distribution of leafwise meromorphic functions. If a lamination has an ergodic harmonic measure, a variant of defect relation in Nevanlinna theory is obtained for almost all leaves. It gives a bound of the number of omitted points by those functions. Consequently we have a Picard type theorem for leafwise meromorphic functions.     

On the groups of c-projective transformations of complete Kähler manifolds

We show that for any complete connected Kähler manifold, the index of the group of complex affine transformations in the group of c-projective transformations is at most two unless the Kähler manifold is isometric to complex projective space equipped with a positive constant multiple of the Fubini–Study metric. This establishes a stronger version of the recently proved Yano–Obata conjecture for complete Kähler manifolds.     

An integral invariant from the view point of locally conformally Kähler geometry

In this article we study an integral invariant which obstructs the existence on a compact complex manifold of a volume form with the determinant of its Ricci form proportional to itself, in particular obstructs the existence of a Kähler-Einstein metric, and has been studied since 1980s. We study this invariant from the view point of locally conformally Kähler geometry. We first see that we can define an integral invariant for coverings of compact complex manifolds with automorphic volume forms. This situation typically occurs for locally conformally Kähler manifolds. Secondly, we see that this invariant coincides with the former one. We also show that the invariant vanishes for any compact Vaisman manifold.     

Compact indefinite almost Kähler Einstein manifolds

In the present article, we extend the integral formula on a compact almost Kähler manifold with positive-definite metric to the one on a compact indefinite almost Kähler manifold and give its applications for some special indefinite almost Kähler Einstein manifolds taking the related problems to the indefinite analogy of Goldberg conjecture into consideration.     

The Kähler Ricci flow on Fano surfaces (I)

Suppose {(M, g(t)), 0 ≤ t < ∞} is a Kähler Ricci flow solution on a Fano surface. If |Rm| is not uniformly bounded along this flow, we can blowup at the maximal curvature points to obtain a limit complete Riemannian manifold X. We show that X must have certain topological and geometric properties. Using these properties, we are able to prove that |Rm| is uniformly bounded along every Kähler Ricci flow on toric Fano surface, whose initial metric has toric symmetry. In particular, such a Kähler Ricci flow must converge to a Kähler Ricci soliton metric. Therefore we give a new Ricci flow proof of the existence of Kähler Ricci soliton metrics on toric Fano surfaces.     

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.     

Generalized hermann theorems and conformal holomorphic submersions

It is shown that if a Kähler manifold admits a holomorphic Riemann submersion, then this manifold is locally reducible. Hermann's well-known theorems are generalized to conformal and holomorphic submersions. A method for constructing Kähler fiber spaces with holomorphic conformal (non-Riemannian) projection and totally geodesic isomorphic fibers is suggested. The method allows us to construct complete, including compact, Kähler fiber spaces of the specified type.     

On the Space of Kähler Potentials

We consider the geodesic equation for the generalized Kähler potential with only mixed second derivatives bounded. We show that given two such generalized Kähler potentials, there is a unique geodesic segment such that for each point on the geodesic, the generalized Kähler potential has uniformly bounded mixed second derivatives (in manifold directions). This generalizes a fundamental theorem of Chen (2000) on the space of Kähler potentials.© 2014 Wiley Periodicals, Inc.     

Holomorphic submersions of locally conformally Kähler manifolds

A locally conformally Kähler (LCK) manifold is a complex manifold covered by a Kähler manifold, with the covering group acting by homotheties. We show that if such a compact manifold \(X\) admits a holomorphic submersion with positive-dimensional fibers at least one of which is of Kähler type, then \(X\) is globally conformally Kähler or biholomorphic, up to finite covers, to a small deformation of a Vaisman manifold (i.e., a mapping torus over a circle, with Sasakian fiber). As a consequence, we show that the product of a compact non-Kähler LCK and a compact Kähler manifold cannot carry a LCK metric.     

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 Kähler surfaces of constant positive scalar curvature

LetM be a compact complex 2-manifold that admits a Kähler metric for which the integral of the scalar curvature is positive. Suppose, moreover, that b₁ (M) > 2. Then, ifM is blown up at sufficiently many points, the resulting complex manifold ~M admits Kähler metrics of constant positive scalar curvature.     

Geometric Schrdinger-Airy Flows on Khler Manifolds

We define a class of geometric flows on a complete Khler manifold to unify some physical and mechanical models such as the motion equations of vortex filament, complex-valued mKdV equations, derivative nonlinear Schrdinger equations etc. Furthermore, we consider the existence for these flows from S~1into a complete Khler manifold and prove some local and global existence results.     

Coupled vortex equations and moduli: deformation theoretic approach and Kähler geometry

We investigate differential geometric aspects of moduli spaces parametrizing solutions of coupled vortex equations over a compact Kähler manifold X. These solutions are known to be related to polystable triples via a Kobayashi–Hitchin type correspondence. Using a characterization of infinitesimal deformations in terms of the cohomology of a certain elliptic double complex, we construct a Hermitian structure on these moduli spaces. This Hermitian structure is proved to be Kähler. The proof involves establishing a fiber integral formula for the Hermitian form. We compute the curvature tensor of this Kähler form. When X is a Riemann surface, the holomorphic bisectional curvature turns out to be semi-positive. It is shown that in the case where X is a smooth complex projective variety, the Kähler form is the Chern form of a Quillen metric on a certain determinant line bundle.     

Einstein almost cokähler manifolds

We study an odd‐dimensional analogue of the Goldberg conjecture for compact Einstein almost Kähler manifolds. We give an explicit non‐compact example of an Einstein almost cokähler manifold that is not cokähler. We prove that compact Einstein almost cokähler manifolds with nonnegative *‐scalar curvature are cokähler (indeed, transversely Calabi–Yau); more generally, we give a lower and upper bound for the *‐scalar curvature in the case that the structure is not cokähler. We prove similar bounds for almost Kähler Einstein manifolds that are not Kähler.     

Almost Kähler metrics of negative scalar curvature on symplectic manifolds

We show that every symplectic manifold of dimension ≥ 4 admits a complete compatible almost Kähler metric of negative scalar curvature. And we discuss the C ⁰-closure of the set of almost Kähler metrics of negative scalar curvature. Some local versions are also proved.     

On the Existence of Proper Nearly Kenmotsu Manifolds

We prove that a nearly Kenmotsu manifold is locally isometric to the warped product of a real line and a nearly Kähler manifold. As consequence, a normal nearly Kenmotsu manifold is Kenmotsu. Furthermore, we show that there do not exist nearly Kenmotsu hypersurfaces of nearly Kähler manifolds.

     

On toric locally conformally Kähler manifolds

We study compact toric strict locally conformally Kähler manifolds. We show that the Kodaira dimension of the underlying complex manifold is \(-\infty \), and that the only compact complex surfaces admitting toric strict locally conformally Kähler metrics are the diagonal Hopf surfaces. We also show that every toric Vaisman manifold has lcK rank 1 and is isomorphic to the mapping torus of an automorphism of a toric compact Sasakian manifold.     

Existence of a complete holomorphic vector field via the Kähler–Einstein metric

In this paper, we study the existence of a complete holomorphic vector field on a strongly pseudoconvex complex manifold admitting a negatively curved complete Kähler–Einstein metric and a discrete sequence of automorphisms. Using the method of potential scaling, we will show that there is a potential function of the Kähler–Einstein metric whose differential has a constant length. Then, we will construct a complete holomorphic vector field from the gradient vector field of the potential function.

     

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.     

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.