Associated families of pluriharmonic maps and isotropy

Like minimal surface immersions in 3-space, pluriharmonic maps into symmetric spaces allow a one-parameter family of isometric deformations rotating the differential (“associated family”); in fact, pluriharmonic maps are characterized by this property. We give a geometric proof of this fact and investigate the “isotropic” case where this family is constant. It turns out that isotropic pluriharmonic maps arise from certain holomorphic maps into flag manifolds. Further, we also consider higher dimensional generalizations of constant mean curvature surfaces which are Kähler submanifolds with parallel (1,1) part of their soecond fundamental form; under certain restrictions there are also characterized by having some kind of (“weak”) associated family. Examples where this family is constant arise from extrinsic Kähler symmetric spaces.     

Minimal surfaces in symmetric spaces with parallel second fundamental form

In this paper, we study geometry of isometric minimal immersions of Riemannian surfaces in a symmetric space by moving frames and prove that the Gaussian curvature must be constant if the immersion is of parallel second fundamental form. In particular, when the surface is \(S^2\), we discuss the special case and obtain a necessary and sufficient condition such that its second fundamental form is parallel. We also consider isometric minimal two-spheres immersed in complex two-dimensional Kähler symmetric spaces with parallel second fundamental form, and prove that the immersion is totally geodesic with constant Kähler angle if it is neither holomorphic nor anti-holomorphic with Kähler angle \(\alpha \ne 0\) (resp. \(\alpha \ne \pi \)) everywhere on \(S^2\).     

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.     

Central Kähler metrics with non-constant central curvature

The central curvature of a Riemannian metric is the determinant of its Ricci endomorphism, while the scalar curvature is its trace. A Kähler metric is called central if the gradient of its central curvature is a holomorphic vector field. Such metrics may be viewed as analogs of the extremal Kähler metrics defined by Calabi. In this work, central metrics of non-constant central curvature are constructed on various ruled surfaces, most notably the first Hirzebruch surface. This is achieved via the momentum construction of Hwang and Singer, a variant of an ansatz employed by Calabi (1979) and by Koiso and Sakane (1986). Non-existence, real-analyticity and positivity properties of central metrics arising in this ansatz are also established.     

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.     

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.     

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.

     

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.     

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.     

Almost Hermitian structures induced from a Kähler structure which has constant holomorphic sectional curvature

We obtain a non-Kähler almost Hermitian manifold of constant holomorphic sectional curvature by changing the almost complex structure in a Kähler manifold of constant holomorphic sectional curvature.

     

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.     

4-dimensional anti-Kähler manifolds and Weyl curvature

On a 4-dimensional anti-Kähler manifold, its zero scalar curvature implies that its Weyl curvature vanishes and vice versa. In particular any 4-dimensional anti-Kähler manifold with zero scalar curvature is flat.     

Log canonical thresholds of certain Fano hypersurfaces

We study log canonical thresholds on quartic threefolds, quintic fourfolds, and double spaces. As an important application, we show that they have Kähler–Einstein metrics if they are general.     

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.     

Codimension of immersions with parallel pluri-mean curvature

Immersions with parallel pluri-mean curvature into euclidean n-space generalize constant mean curvature immersions of surfaces to Kähler manifolds of complex dimension m. Examples are the standard embeddings of Kähler symmetric spaces into the Lie algebra of its transvection group. We give a lower bound for the codimension of arbitrary ppmc immersions. In particular we show that M is locally symmetric if the codimension is minimal.     

On projective invariants of the complex Finsler spaces

In this paper we extend the results on projective changes of complex Finsler metrics obtained in Aldea and Munteanu (2012) [3], by the study of projective curvature invariants of a complex Finsler space. By means of these invariants, the notion of complex Douglas space is then defined. A special approach is devoted to the obtaining of equivalence conditions for a complex Finsler space to be a Douglas one. It is shown that any weakly Kähler Douglas space is a complex Berwald space. A projective curvature invariant of Weyl type characterizes complex Berwald spaces. These must be either purely Hermitian of constant holomorphic curvature, or non-purely Hermitian of vanishing holomorphic curvature. Locally projectively flat complex Finsler metrics are also studied.     

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.     

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.     

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.