ON A KHLER VERSION OF CHEEGER-GROMOLL-PERELMAN'S SOUL THEOREM

In this note, we will prove a Khler version of Cheeger-Gromoll-Perelman's soul theorem, only assuming the sectional curvature is nonnegative and bisectional curvature is positive at one point.     

Characterizations of Null Holomorphic Sectional Curvature of GCR-Lightlike Submanifolds of Indefinite Nearly Khler Manifolds

We obtain the expressions for sectional curvature, holomorphic sectional curvature and holomorphic bisectional curvature of a GCR-lightlike submanifold of an indefinite nearly Khler manifold and obtain characterization theorems for holomorphic sectional and holomorphic bisectional curvature. We also establish a condition for a GCR-lightlike submanifold of an indefinite complex space form to be a null holomorphically flat.     

An Ohsawa-Takegoshi theorem on compact Khler manifolds

We prove two extension theorems of Ohsawa-Takegoshi type on compact Khler manifolds.In our proof,there are many complications arising from the regularization process of quasi-psh functions on compact Khler manifolds,and unfortunately we only obtain particular cases of the expected result.We remark that the two special cases we proved are natural,since they occur in many situations.We hope that the new techniques we develop here will allow us to obtain the general extension result of Ohsawa-Takegoshi type on compact Khler manifolds in a near future.     

Affinely equivalent Khler-Finsler metrics on a complex manifold

The purpose of the present paper is to investigate affinely equivalent Khler-Finsler metrics on a complex manifold.We give two facts (1) Projectively equivalent Khler-Finsler metrics must be affinely equivalent;(2) a Khler-Finsler metric is a Khler-Berwald metric if and only if it is affinely equivalent to a Khler metric.Furthermore,we give a formula to describe the affine equivalence of two weakly Khler-Finsler metrics.     

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.     

On almost complex curves and Hopf hypersurfaces in the nearly Khler six-sphere

We study the almost complex curves and Hopf hypersurfaces in the nearly Khler S6(1),and their relations.For Hopf hypersurfaces,we give a classification theorem under some additional conditions.For compact almost complex curves,we obtain some interesting global results with respect to Gaussian curvature,area and the genus.     

Conjugate Points on a Type of Khler Manifolds

We study conjugate points on a type of Khler manifolds, which are submanifolds of Grassmannian manifolds. And then we give the applications to the study of the index of geodesics and homotopy groups.     

Sasakian空间形式中子流形的全实截面曲率(英文)

By using the methods introduced by Chen[Chen Bang-yen,A series of Ka¨hlerian invarianrts and their applications to Khlerian geometry,Beitrge Algebra Geom,2001,42（1）：165-178],we establish some inequalities for invariant submanifolds in a Sasakian space form involving totally real sectional curvature and the scalar curvature.Moreover,we consider the case of equalities.     

Holomorphic Lefschetz Fixed Point Formula for Non-compact Khler Manifolds

The authors obtain a holomorphic Lefschetz fixed point formula for certain non-compact "hyperbolic" Khler manifolds (e.g. Khler hyperbolic manifolds, bounded domains of holomorphy) by using the Bergman kernel. This result generalizes the early work of Donnelly and Fefferman.     

Chern-Ricci curvatures,holomorphic sectional curvature and Hermitian metrics

We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics.We prove that a compact locally conformal K?hler manifold with the constant nonpositive holomorphic sectional curvature is K?hler.We also give examples of complete non-K?hler metrics with pointwise negative constant but not globally constant holomorphic sectional curvature,and complete non-K?hler metrics with zero holomorphic sectional curvature and nonvanishing curvature tensors.     

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.     

Construction of homogeneous minimal 2-spheres in complex Grassmannians

In this paper, we construct a class of homogeneous minimal 2-spheres in complex Grassmann manifolds by applying the irreducible unitary representations of SU (2). Furthermore, we compute induced metrics, Gaussian curvatures, Khler angles and the square lengths of the second fundamental forms of these homogeneous minimal 2-spheres in G(2, n + 1) by making use of Veronese sequence.     

A NOTE ON CONICAL KHLER-RICCI FLOW ON MINIMAL ELLIPTIC KHLER SURFACES

We prove that, under a semi-ampleness type assumption on the twisted canonical line bundle, the conical Khler-Ricci flow on a minimal elliptic Khler surface converges in the sense of currents to a generalized conical Khler-Einstein on its canonical model. Moreover,the convergence takes place smoothly outside the singular fibers and the chosen divisor.     

Conformal minimal two-spheres in Q_n

In this paper we study,using moving frames,conformal minimal two-spheres S2 immersed into a complex hyperquadric Qn equipped with the induced Fubini-Study metric from a complex projective n+1-space CPn+1.Two associated functions τX and τY are introduced to classify the problem into several cases.It is proved that τX or τY must be identically zero if f:S2 → Qn is a conformal minimal immersion.Both the Gaussian curvature K and the Khler angle θ are constant if the conformal immersion is totally geodesic.It i...     

Regularity Properties of the Degenerate Monge-Ampère Equations on Compact Khler Manifolds

The author establishes a result concerning the regularity properties of the degenerate complex Monge-Ampère equations on compact Khler manifolds.     

Isotropic Lagrangian submanifolds in the homogeneous nearly Khler S~3× S~3

We show that isotropic Lagrangian submanifolds in a 6-dimensional strict nearly Khler manifold are totally geodesic. Moreover, under some weaker conditions, a complete classification of the J-isotropic Lagrangian submanifolds in the homogeneous nearly Khler S~3× S~3 is also obtained. Here, a Lagrangian submanifold is called J-isotropic, if there exists a function λ, such that g((▽h)(v, v, v), J v) = λ holds for all unit tangent vector v.     

On the CR Poincaré–Lelong Equation,Yamabe Steady Solitons and Structures of Complete Noncompact Sasakian Manifolds

In this paper, we solve the so-called CR Poincaré–Lelong equation by solving the CR Poisson equation on a complete noncompact CR(2n + 1)-manifold with nonegative pseudohermitian bisectional curvature tensors and vanishing torsion which is an odd dimensional counterpart of K?hler geometry. With applications of this solution plus the CR Liouvelle property, we study the structures of complete noncompact Sasakian manifolds and CR Yamabe steady solitons.     

Canonical Metrics on Generalized Cartan-Hartogs Domains

In this paper, the author considers a class of bounded pseudoconvex domains,i.e., the generalized Cartan-Hartogs domains Ω(μ, m). The first result is that the natural Khler metric g~(Ω(μ,m)) of Ω(μ, m) is extremal if and only if its scalar curvature is a constant. The second result is that the Bergman metric, the Ka¨hler-Einstein metric, the Carathéodary metric, and the Koboyashi metric are equivalent for Ω(μ, m).     

RIGIDITY THEOREMS OF COMPLETE K?HLER-EINSTEIN MANIFOLDS AND COMPLEX SPACE FORMS

We concentrate on using the traceless Ricci tensor and the Bochner curvature tensor to study the rigidity problems for complete K?hler manifolds. We derive some elliptic differential inequalities from Weitzenb?ck formulas for the traceless Ricci tensor of K?hler manifolds with constant scalar curvature and the Bochner tensor of K?hler-Einstein manifolds respectively. Using elliptic estimates and maximum principle, several L~p and L~∞ pinching results are established to characterize K?hler-Einstein manifolds among K?hler manifolds with constant scalar curvature and complex space forms among K?hler-Einstein manifolds.Our results can be regarded as a complex analogues to the rigidity results for Riemannian manifolds. Moreover, our main results especially establish the rigidity theorems for complete noncompact K?hler manifolds and noncompact K?hler-Einstein manifolds under some pointwise pinching conditions or global integral pinching conditions. To the best of our knowledge,these kinds of results have not been reported.     

本刊英文版Vol.27(2011),No.8论文摘要

<正>Schrdinger Soliton from Lorentzian Manifolds Chong SONG You De WANG Abstract In this paper,we introduce a new notion named as Schrdinger soliton.The socalled Schrdinger solitons are a class of solitary wave solutions to the Schrdinger flow equation from a Riemannian manifold or a Lorentzian manifold M into a Khler manifold N.If the target manifold N admits a Killing potential,then the Schrdinger soliton reduces to a harmonic