Quasi—conformally Flat Manifolds with Constant Scalar Curvature

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Rigidity Theorems for Complete Sasakian Manifolds with Constant Pseudo-Hermitian Scalar Curvature

The orthogonal decomposition of the Webster curvature provides us a way to characterize some canonical metrics on a pseudo-Hermitian manifold. We derive some subelliptic differential inequalities from the Weitzenböck formulas for the traceless pseudo-Hermitian Ricci tensor of Sasakian manifolds with constant pseudo-Hermitian scalar curvature and the Chern–Moser tensor of the Sasakian pseudo-Einstein manifolds, respectively. By means of either subelliptic estimates or maximum principle, some rigidity theorems are established to characterize Sasakian pseudo-Einstein manifolds among Sasakian manifolds with constant pseudo-Hermitian scalar curvature and Sasakian space forms among Sasakian pseudo-Einstein manifolds, respectively.     

Finsler Manifolds with Positive Constant Flag Curvature

It is shown that a Finsler metric with positive constant flag curvature and vanishing mean tangent curvature must be Riemannian. As applications, we also discuss the case of Cheng's maximal diameter theorem and Green's maximal conjugate radius theorem in Finsler manifolds.     

Pluriclosed Manifolds with Constant Holomorphic Sectional Curvature

A long-standing conjecture in complex geometry says that a compact Hermitian manifold with constant holomorphic sectional curvature must be Kèahler when the constant is non-zero and must be Chern flat when the constant is zero. The conjecture is known in complex dimension 2 by the work of Balas-Gauduchon in 1985(when the constant is zero or negative) and by Apostolov±Davidov±Muskarov in 1996(when the constant is positive). For higher dimensions, the conjecture is still largely unknown. In this a...     

Deformations of Almost-Kähler Metrics with Constant Scalar Curvature on Compact Kähler Manifolds

We prove the existence of infinite-dimensional families of(non-Kähler) almost-Kähler metrics with constant scalar curvature oncertain compact manifolds. These are obtained by deformingconstant-scalar-curvature Kähler metrics on suitable compact complexmanifolds. We prove several other similar results concerning the scalarcurvature and/or the *-scalar curvature. We also discuss thescalar curvature functions of almost-Kähler metrics.     

COMPLETE HYPERSURFACES WITH CONSTANT SCALAR CURVATURE AND CONSTANT MEAN CURVATURE IN R~4

Let M be a 3-dimersional complete and connected hypersurface immersed in R~4. If thescalar curvature R and the mean curvature |H| of M are constants, where |H|≠0, R≥0,then there are only three cases: R=6|H|~2, 9/2|H|~2 and 0. Moreovon we can find somehypersurfaces appropriate to these cases.     

负曲率流形上给定数量曲率的共形形变

本文研究具强负曲率Cartan－Hadamard流形M~n（n≥3）上给定数量曲率函数S的共形形变问题．利用上下解方法,并通过精心构造上解,我们获得了当完备的共形形变度量存在时,函数S在无穷远附近的最佳渐近性态．在较一般情况下,我们还给出了共形数量曲率方程解的渐近估计．     

空间形式中常数量率的子流形

本文利用一个类似于Cheng和Yau引进的微分算子的新微分算子，得到了单位球面中常数量率的紧致子流形的一个刚性结果．     

Dirac Operators on Hypersurfaces of Manifolds with Negative Scalar Curvature

We give a sharp extrinsic lower bound for the first eigenvaluesof the intrinsic Dirac operator of certain hypersurfaces boundinga compact domain in a spin manifold of negative scalar curvature.Limiting-cases are characterized by the existence, on the domain,of imaginary Killing spinors. Some geometrical applications, as anAlexandrov type theorem, are given.     

Unit Tangent Sphere Bundles with Constant Scalar Curvature

As a first step in the search for curvature homogeneous unit tangent sphere bundles we derive necessary and sufficient conditions for a manifold to have a unit tangent sphere bundle with constant scalar curvature. We give complete classifications for low dimensions and for conformally flat manifolds. Further, we determine when the unit tangent sphere bundle is Einstein or Ricci-parallel.     

Best Sobolev Constants and Manifolds with Positive Scalar Curvature Metrics

We study the Yamabe invariant of manifolds which admit metrics of positive scalar curvature. Analysing `best Sobolev constants'we give a technique to find positive lower bounds for the invariant.We apply these ideas to show that for any compact Riemannian manifold (N ⁿ ,g) of positive scalarcurvature there is a positive constant K =K(N, g), which depends only on (N, g), such that for any compact manifold M ^m , the Yamabe invariantof M ^m × N ⁿ is no less than K times the invariant ofS ^{n + m} . We will find some estimates for the constant K in the case N =S ⁿ .     

Scalar Curvature Rigidity for Asymptotically Locally Hyperbolic Manifolds

Rigidity results for asymptotically locally hyperbolic manifolds with lower bounds on scalar curvature are proved using spinor methods related to the Witten proof of the positive mass theorem. The argument is based on a study of the Dirac operator defined with respect to the Killing connection. The existence of asymptotic Killing spinors is related to the spin structure on the end. The expression for the mass is calculated and proven to vanish for conformally compact Einstein manifolds with conformal boundary a spherical space form, giving rigidity. In the four dimensional case, the signature of the manifold is related to the spin structure on the end and explicit formulas for the relevant invariants are given.     

Complete Hypersurfaces with Constant Laguerre Scalar Curvature in R~n

Let x:M~(n-1)→R~n be an umbilical free hypersurface with non-zero principal curvatures.Two basic invariants of M under the Laguerre transformation group of R~n are Laguerre form C and Laguerre tensor L.In this paper,we prove the following theorem:Let M be an(n-1)-dimensional(n 3) complete hypersurface with vanishing Laguerre form and with constant Laguerre scalar curvature R in R~n,denote the trace-free Laguerre tensor by L =L-1/n-1tr(L)·Id.If sup_M ||L||=0,then M is Laguerre equivalent to a Laguerre isotropic hypersurface;and if sup_M ||L||=((n-1)(n-2))~(1/2)R/((n-1)(n-2)(n-3)),M is Laguerre equivalent to the hypersurface x:H~1×S~(n-2)→R~n.     

A Characterization of Quadric Constant Scalar Curvature Hypersurfaces of Spheres

Let M?S ⁴ be a complete orientable hypersurface with constant scalar curvature. For any v∈R ⁵, let us define the two real functions \(l_{v}, f_{v}:M\to{\bf R}\) on M by l _v (x)=〈x,v〉 and f _v (x)=〈ν(x),v〉 with ν:M→S ⁴ a Gauss map of M. In this paper, we show that if we have that l _v =λf _v for some nonzero vector v∈R ⁵ and some real number λ, then M is either totally umbilical (a Euclidean sphere) or M is a Cartesian product of Euclidean spheres. We will also show with an example that the completeness condition is needed in this theorem.     

Constant Scalar Curvature Metrics on Toric Surfaces

The main result of the paper is an existence theorem for a constant scalar curvature Kahler metric on a toric surface, assuming the K-stability of the manifold. The proof builds on earlier papers by the author, which reduce the problem to certain a priori estimates. These estimates are obtained using a combination of arguments from Riemannian geometry and convex analysis. The last part of the paper contains a discussion of the phenomena that can be expected when the K-stability does not hold and solutions do not exist. Received: May 2008, Revision: December 2008, Accepted: December 2008     

Examples of Hermitian Manifolds with Pointwise Constant Holomorphic Sectional Curvature

We construct non-compact examples of Hermitian manifolds with pointwise constant holomorphic sectional curvature. Our examples are obtained by conformal change of the metric on an open set of the complex space form.     

关于具有常平均曲率和数量曲率超曲面的Moebius几何的一个注记

设x：M→S^n＋1（n≥23）是n＋1-维单位球中的无脐点超曲面，Moebius不变量G，Ф，A和B分别表示x的Moebius度量，Moebius形式，Blaschke形式和Moebius第二基本形式．本文证明了如果x的Moebius形式圣平行，并且A＋λG＋μB=0，其中λ，μ分别是定义在M上的光滑函数，那么Ф=0，由此及李海中、王长平（2003年）文献中的分类定理给出了眇州中具有平行的Moebius形式及满足A＋λG＋μB=0的超曲面的分类．此结果推广了他们及张廷枋（2003年）文献中的结果．     

复射影空间中具有常数量曲率的实超曲面

将球面上常数量曲率超曲面推广到复射影空间中,得到此类实超曲面的某些积分不等式.     

Minimal Submanifolds in Sp+3 with Constant Scalar Curvature

Let M be a compact, minimal 3-dimensional submanifold with constant scalar curvature R immersed in the standard sphere S^3+p. In codimension 1, we know from the work that has been done on Chern’s conjecture that M is isoparametric and R = 3D0, R = 3D3 or R = 3D6. In this paper we extend this result from codimension one to compact submanifolds with a flat normal bundle and give a complete classification.