A Theorem of Liouville Type for p-Harmonic Morphisms

In this article we prove a Liouville type theorem for p-harmonic morphisms. We show that if : MNis a p-harmonic morphism (p2) from a complete noncompact Riemannian manifold Mof nonnegative Ricci curvature into a Riemannian manifold Nof nonpositive scalar curvature such that the p-energy E _p (), or (2p–2)-energy E _2p–2() is finite, then is constant.     

A Nonexistence Theorem of Proper Harmonic Morphisms Between Hyperbolic Spaces

We provide a nonexistence theorem of harmonic morphisms between hyperbolic spaces which are proper and C ²-smooth to the boundary.     

一类非线性椭圆型方程的一个Liouville定理

设 (Mn,g)是一个 n维的完备黎曼流形 ,其 Ricci曲率满足 Ric M(x)≥ - A(1 r2 (x) ln2 (2 r(x) ) ) ,其中 A是非负常数 ,r(x)表示点 x∈ M到某固定点 x0 ∈ M的测地距离 .则 M上方程 Δu Su Kuα=0在下述条件“ (i)在 M上 S≤ 0 ;(ii)在 M上 K<0且有常数 a>0使在一个紧集之外 K≤ - a2 ;(iii)常数 α>1”下的 C2 -非负解只有零解 .     

关于调和映照的一个Liouville型定理

令M、N是完备Remann流形,设M上不存在任何非平凡的有界调和函数,N的截面曲率具有上界其为k＞0.设uM→N是一个调和映照且u(M)∈BR(P),其中R＝·如果BR(P)位于P的割迹之内,并且μ（M）∩эBR(P)最多只有一个点,则u必是一个常值映照.     

Harmonic polynomial morphisms between Euclidean spaces

Two non-existence theorems on harmonic polynomial morphisms between Euclidean spaces have been shown. Project supported partially by the National Natural Science Foundation of China (Grant No. 19531050) and the State Education Commission Foundation of China.     

微分形式的Liouville型定理

本文建立了具有有界的负截面曲率的完备单连通黎曼流形上 .其应力能量张量守恒的 L2 -形式的一个消失定理 .从而推广了忻元龙的新近结果 ,给出了 Dodziuk猜想的部分回答     

Non-existence for Harmonic Maps on Complete Noncompact Manifolds

In this paper, we prove a nonexistence theorem on harmonic maps. This generalizes the well-known Liouville-type theorem on harmonic maps due to S.Y. Cheng and H.I Choi.     

一类完备Riemann流形的正调和函数

本文主要讨论一类完备Ｒｉｅｍａｎｎ流形上的调和函数所组成的线性空间．推广了Ｐ．Ｌｉ及Ｌ．Ｆ．Ｔａｍ^{[5], [7]}和和Ｇｒｅｅｎｅ－Ｗｕ^[3]中的结果．     

A New Proof of Calabi-Yau's Theorem

We give a new proof of Calabi-Yau's theorem on the volume growth of Rie- mannian manifolds with non-negative Ricci curvature.     

A characterization of the Clifford torus

In this paper, we prove that an -dimensional closed minimal hypersurface with Ricci curvature of a unit sphere is isometric to a Clifford torus if , where is the squared norm of the second fundamental form of .

     

A New Proof of Calabi-Yau＇s Theorem

We give a new proof of Calabi-Yau＇s theorem on the volume growth of Riemannian manifolds with non-negative Ricci curvature.     

Categories of Projective Geometries with Morphisms and Homomorphisms

Projective geometries studied as Pasch geometries possess morphisms and homomorphisms. A homomorphic image of a projective geometry is shown to be projective. A projective geometry is shown to be Desarguesian iff it is a homomorphic image of a higher dimensional one, which in a sense is dual to the classical imbedding theorem. Semi-linear maps induce morphisms which are homomorphisms iff the associated homomorphisms of skewfields are isomorphisms. Projective geometries form categories with morphisms as well as homomorphisms and Desarguesian ones form a subcategory with Desarguesian homomorphisms.     

A Liouville Theorem for Matrix-Valued Harmonic Functions on Nilpotent Groups

Let be a non-degenerate positive M_n-valued measure on a locallycompact group G with |||| = 1. An M_n-valued Borel function fon G is called -harmonic if for all x G. Given such a function f which is bounded and leftuniformly continuous on G, it is shown that every central elementin G is a period of f. Further, it is shown that f is constantif G is nilpotent or central. 2000 Mathematics Subject Classification31C05, 43A05, 45E10, 46G10.     

On a sharp volume estimate for gradient Ricci solitons with scalar curvature bounded below

In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota’s argument we obtain a local lower bound estimate of the scalar curvature for the Ricci flow on complete manifolds. Consequently, one has a sharp estimate of the scalar curvature for expanding Ricci solitons; we also provide a direct (elliptic) proof of this sharp estimate. Moreover, if the scalar curvature attains its minimum value at some point, then the manifold is Einstein.     

关于局部对称空间中极小子流形的一个Ricci曲率pinching定理

研究局部对称空间中具有正Ricci曲率的完备极小子流形,得到了关于子流形Ricci曲率的一个pinching定理,把Norio Ejiri的结论从外围空间为球空间推广到局部对称空间中。     

一类流形的刚性定理

本文通过具有良好性质的子流形的存在性,证明了一类流形的一个刚性定理,并得到形如Ｂｏｎｎｅｔ－Ｍｙｅｒｓ定理的推论．我们还指出,在主要定理中全测地子流形的条件一般不能减弱为极小子流形．     

Ridigity of Ricci solitons with weakly harmonic Weyl tensors

In this paper, we prove rigidity results on gradient shrinking or steady Ricci solitons with weakly harmonic Weyl curvature tensors. Let be a compact gradient shrinking Ricci soliton satisfying with constant. We show that if satisfies , then is Einstein. Here denotes the Weyl curvature tensor. In the case of noncompact, if M is complete and satisfies the same condition, then M is rigid in the sense that M is given by a quotient of product of an Einstein manifold with Euclidean space. These are generalizations of the previous known results in 10 , 14 and 19 . Finally, we prove that if is a complete noncompact gradient steady Ricci soliton satisfying , and if the scalar curvature attains its maximum at some point in the interior of M, then either is flat or isometric to a Bryant Ricci soliton. The final result can be considered as a generalization of main result in 3 .     

Generalized Ejiri's Rigidity Theorem for Submanifolds in Pinched Manifolds

Let $M^{n}(n\geq4)$ be an oriented compact submanifold with parallel mean curvature in an $(n+p)$-dimensional complete simply connected Riemannian manifold $N^{n+p}$. Then there exists a constant $\delta(n,p)\in(0,1)$ such that if the sectional curvature of $N$ satisfies $\ov{K}_{N}\in[\delta(n,p), 1]$, and if $M$ has a lower bound for Ricci curvature and an upper bound for scalar curvature, then $N$ is isometric to $S^{n+p}$. Moreover, $M$ is either a totally umbilic sphere $S^n\big(\frac{1}{\sqrt{1+H^2}}\big)$, a Clifford hypersurface $S^{m}\big(\frac{1}{\sqrt{2(1+H^2)}}\big)\times S^{m}\big(\frac{1}{\sqrt{2(1+H^2)}}\big)$ in the totally umbilic sphere $S^{n+1}\big(\frac{1}{\sqrt{1+H^2}}\big)$ with $n=2m$, or $\mathbb{C}P^{2}\big(\frac{4}{3}(1+H^2)\big)$ in $S^7\big(\frac{1}{\sqrt{1+H^2}}\big)$. This is a generalization of Ejiri''s rigidity theorem.     

局部对称共形平坦黎曼流形中紧致极小子流形的一个刚性定理

本文研究局部对称共形平坦黎曼流形中紧致极小子流形,得到了这类子流形第二基本形式模长平方关于外围空间Ricci曲率的—个拼挤定理,推广了文[1]中的结果.