DEFORMING METRICS WITH POSITIVE CURVATURE BY A FULLY NONLINEAR FLOW

By studying a fully nonlinear flow deforming conformal metrics on a compact and connected manifold, we prove the long time existence and the exponential convergence of the solutions of the flow for any initial metric g0 with the Schouten tensor Ag0 ∈ Γk.

Symmetrically Harmonic Kaluza-Klein Metrics on Tangent Bundles

Let (M, g) be a Riemannian manifold and G be a Kaluza-Klein metric on its tangent bundle TM. A metric H on TM is said to be symmetrically harmonic to G if the metrics G and H are harmonic w.r.t. each other;that is the identity maps id: (TM,G) → (TM,H) and id: (TM,H) → (TM,G) are both harmonic maps. In this work we study Kaluza-Klein metrics H on TM which are symmetrically harmonic to G. In particular, we characterize and determine horizontally and vertically conformal Kaluza-Klein metrics H on TM, which are symmetrically harmonic to G.

Locally Conformal Pseudo-Kahler Finsler Manifolds

In this paper,we give a necessary and sucient condition for a strongly pseudoconvex complex Finsler metric to be locally conformal pseudo-Kahler Finsler.As an application,we nd any complete strongly convex and locally conformal pseudo-Kahler Finsler manifold,which is simply connected or whose fundamental group contains elements of nite order only,can be given a Kahler metric.

曲面上的曲率在理论物理中的一些应用

In this survey article,we present two applications of surface curvatures in theoretical physics.The first application arises from biophysics in the study of the shape of cell vesicles involving the minimization of a mean curvature type energy called the Helfrich bending energy.In this formalism,the equilibrium shape of a cell vesicle may present itself in a rich variety of geometric and topological characteristics.We first show that there is an obstruction,arising from the spontaneous curvature,to the existence of a minimizer of the Helfrich energy over the set of embedded ring tori.We then propose a scale-invariant anisotropic bending energy,which extends the Canham energy,and show that it possesses a unique toroidal energy minimizer,up to rescaling,in all parameter regime.Furthermore,we establish some genus-dependent topological lower and upper bounds,which are known to be lacking with the Helfrich energy,for the proposed energy.We also present the shape equation in our context,which extends the Helfrich shape equation.The second application arises from astrophysics in the search for a mechanism for matter accretion in the early universe in the context of cosmic strings.In this formalism,gravitation may simply be stored over a two-surface so that the Einstein tensor is given in terms of the Gauss curvature of the surface which relates itself directly to the Hamiltonian energy density of the matter sector.This setting provides a lucid exhibition of the interplay of the underlying geometry,matter energy,and topological characterization of the system.In both areas of applications,we encounter highly challenging nonlinear partial differential equation problems.We demonstrate that studies on these equations help us to gain understanding of the theoretical physics problems considered.

THE VALUE DISTRIBUTION OF GAUSS MAPS OF IMMERSED HARMONIC SURFACES WITH RAMIFICATION

Motivated by the result of Chen-Liu-Ru[1],we investigate the value distribution properties for the generalized Gauss maps of weakly complete harmonic surfaces immersed in R^(n) with ramification,which can be seen as a generalization of the results in the case of the minimal surfaces.In addition,we give an estimate of the Gauss curvature for the K-quasiconfomal harmonic surfaces whose generalized Gauss map is ramified over a set of hyperplanes.

Locally Conformal K?hler and Hermitian Yang-Mills Metrics

The author shows that if a locally conformal K?hler metric is Hermitian YangMills with respect to itself with Einstein constant c≤0,then it is a Kahler-Einstein metric.In the case of c>0,some identities on torsions and an inequality on the second Chern number are derived.

On conformal complex Finsler metrics

In this paper,we study conformal transformations in complex Finsler geometry.We first prove that two weakly Kahler Finsler metrics cannot be conformal.Moreover,we give a necessary and sufficient condition for a strongly pseudoconvex complex Finsler metric to be locally conformal weakly Kahler Finsler.Finally,we discuss conformal transformations of a strongly pseudoconvex complex Finsler metric,which preserve the geodesics,holomorphic S-curvatures and mean Landsberg tensors.

Time-Like Conformal Homogeneous Hypersurfaces with Three Distinct Principal Curvatures

A hypersurface x(M)in Lorentzian space R41 is called conformal homogeneous,if for any two points p,q on M,there exists,a conformal transformation of R41,such that(x(M))=x(M),(x(p))=x(q).In this paper,the authors give a complete classifica-tion for regular time-like conformal homogeneous hypersurfaces in R41 with three distinct principal curvatures.

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 Kähler manifold with the constant nonpositive holomorphic sectional curvature is K?hler.We also give examples of complete non-Kähler metrics with pointwise negative constant but not globally constant holomorphic sectional curvature,and complete non-Kähler metrics with zero holomorphic sectional curvature and nonvanishing curvature tensors.

Regular Space-like Hypersurfaces in S1^m+1 with Parallel Para-Blaschke Tensors

In this paper, we give a complete conformal classification of the regular space-like hyper- surfaces in the de Sitter Space S~＋1 with parallel para-Blaschke tensors.

On the Curvature Conjecture of Hua Loo-Keng

It is proved that both the holomorphic sectional and the bisectional curvatures of the conformal Bergman metric

Rigidity of the Hexagonal Delaunay Triangulated Plane

We show the rigidity of the hexagonal Delaunay triangulated plane under Luo’s PL conformality.As a consequence,we obtain a rigidity theorem for a particular type of locally finite convex ideal hyperbolic polyhedra.