常曲率空间中具正Ricci曲率的子流形

设S~(n+p)(1)是一单位球面,M~n是浸入S~(n+p)(1)的具有非零平行平均曲率向量的n维紧致子流形.证明了当n≥4,p≥2时,如果M~n的Ricci曲率不小于(n-2)(1+H~2),则M~n是全脐的或者M~n的Ricci曲率等于(n-2)(1+H~2),进而M~n的几何分类被完全给出.     

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.     

一类浸入在半黎曼卷积空间的完备超曲面

We deal with complete hypersurfaces immersed in a semi-Riemannian warped product of the type eI×f M~n,where M~n is a connected n-dimensional oriented Riemannian manifold.When the fiber M~n is complete with sectional curvature-k≤K_M for some positive constant k,under appropriate restrictions on the norm of the gradient of the height function h,we proceed with our technique in order to guarantee that complete hypersurface immersed in a semi-Riemannian warped product is a slice.Our approach is based on the well known generalized maximum principle and another suitable maximum principle at the infinity due to Yau.     

非齐性广义Morrey空间上双线性强奇异Calder\''{o}n-Zygmund算子及交换子

本文的主要建立非齐性度量测度空间上双线性强奇异积分算子$\widetilde{T}$及交换子$\widetilde{T}_{b_{1},b_{2}}$在广义Morrey空间$M^{u}_{p}(\mu)$上的有界性. 在假设Lebesgue可测函数$u, u_{1}, u_{2}\in\mathbb{W}_{\tau}$, $u_{1}u_{2}=u$,且$\tau\in(0,2)$. 证明了算子$\widetilde{T}$是从乘积空间$M^{u_{1}}_{p_{1}}(\mu)\times M^{u_{2}}_{p_{2}}(\mu)$到空间$M^{u}_{p}(\mu)$有界的, 也是从乘积空间$M^{u_{1}}_{p_{1}}(\mu)\times M^{u_{2}}_{p_{2}}(\mu)$到广义弱Morrey空间$WM^{u}_{p}(\mu)$有界的,其中$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$及$1相似文献   

ON TFE THIRD CONJECTURE OF K.OGIUE

In this paper, the authors prove following result:Let M~n be a complete Bechner-Kaehler submanifold of complex dimension (n≥4) in a complex projective space CP~(n p)(1) of complex dimension n p, endowed with the FubiniStudy metric of constant holomorphic sectional curvature 1. If the sectional curvature K of M~n satisfies K<1, then codimension p of M~n is not less then n(n 1)/2.     

双曲空间中完备子流形的特征值估计

研究了$(n+p)$维双曲空间$\mathbb{H}^{n+p}$中完备非紧子流形的第一特征值的上界.特别地,证明了$\mathbb{H}^{n+p}$中具有平行平均曲率向量$H$和无迹第二基本形式有限$L^q(q\geq n)$范数的完备子流形的第一特征值不超过$\frac{(n-1)^2(1-|H|^2)}{4}$,和$\mathbb{H}^{n+1}(n\leq5)$中具有常平均曲率向量$H$和无迹第二基本形式有限$L^q(2(1-\sqrt{\frac{2}{n}})相似文献   

MINIMAL SUBMANIFOLDS IN A RIEMANNIAN MANIFOLD OF QUASI CONSTANT CURVATURE

A Riemannian manifold V~m which admits isometric imbedding into two spaces V~(m+p)ofdifferent constant curvatures is called a manifold of quasi constant curvature.TheRiemannian curvature of V~m is expressible in the formand conversely.In this paper it is proved that if M~n is any compact minimal submanifoldwithout boundary in a Riemannian manifold V~(n+p)of quasi constant curvature,then∫_(M~u)(2-1/p)σ~2-[na+1/2(b-丨b丨)(n+1)]σ+n(n-1)b~2*丨≥0,where σ is the square of the norm of the second fundamental form of M~n When V~(n+p)is amanifold of constant curvature,b=0,the above inequality reduces to that of Simons.     

ON A CONJECTURE OF K. OGIUE FOR KAEHLER HYPERSURFACES

ＯＮ Ａ ＣＯＮＪＥＣＴＵＲＥ ＯＦ Ｋ． ＯＧＩＵＥ ＦＯＲ ＫＡＥＨＬＥＲ ＨＹＰＥＲＳＵＲＦＡＣＥＳ￥ＳＨＡＮＧＷＥＩＭＩＮＧＡｂｓｔｒａｃｔ：ＡｎａｆｆｉｒｍａｔｉｖｅａｎｓｗｅｒｔｏａｃｏｎｊｅｃｔｕｒｅｏｆＫ．Ｏｇｉｕｅｆｏｒｍｕｌａｔｅ...     

A Remark on the Norm of Integer Order Favard Spaces

For a generator $A$ of a $C_0$-semigroup $T(\cdot)$ on a Banach space $X$ we consider the semi-norm $M^{k}_x:=\limsup_{t\to 0+}\|t^{-1}(T(t)-I)A^{k-1}x\|$ on the Favard space ${\cal F}_{k}$ of order $k$ associated with $A$. The use of this semi-norm is motivated by the functional analytic treatment of time-discretization methods of linear evolution equations. We show that sharp inequalities for bounded linear operators on ${\cal D}(A^k)$ can be extended to the larger space ${\cal F}_{k}$ by using the semi-norm $M^{k}_{(\cdot)}$. We also show that $M^{k}_{(\cdot)}$ is a norm equivalent to the norms that are usually considered in the literature if A has a bounded inverse.     

满足$\varphi l=l\varphi$的3-维仿切触度量流形的分类

Let $M^{3}$ be a 3-dimensional paracontact metric manifold. Firstly, a classification of $M^{3}$ satisfying $\varphi Q=Q\varphi$ is given. Secondly, manifold $M^{3}$ satisfying $\varphi l=l\varphi$ and having $\eta$-parallel Ricci tensor or cyclic $\eta$-parallel Ricci tensor is studied.     

分数次积分交换子在非齐Morrey空间上的紧性

The aim of this paper is to establish the necessary and sufficient conditions for the compactness of fractional integral commutator[b,I_γ]which is generated by fractional integral I_γand function b∈Lip_β(μ)on Morrey space over non-homogeneous metric measure space,which satisfies the geometrically doubling and upper doubling conditions in the sense of Hytonen.Under assumption that the dominating functionλsatisfies weak reverse doubling condition,the author proves that the commutator[b,I_γ]is compact from Morrey space M_q^p(μ)into Morrey space M_t^s(μ)if and only if b∈Lip_β(μ).     

Small Covers over a Product Space

A small cover is a closed manifold $M^{n}$ with a locally standard $(\Bbb{Z}_{2})^{n}$-action such that its orbit space is a simple convex polytope $P^{n}$. Let $\Delta^{n}$ denote an $n$-simplex and $P(m)$ an $m$-gon. This paper gives formulas for calculating the number of D-J equivalent classes and equivariant homeomorphism classes of orientable small covers over the product space $\Delta^{n_1}\times \Delta^{n_2} \times P(m)$, where $n_1$ is odd.     

具有调和曲率与有限$L^p$曲率的完备流形

Let(M~n, g)(n ≥ 3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R and R?m the scalar curvature and the trace-free Riemannian curvature tensor of M, respectively. The main result of this paper states that R?m goes to zero uniformly at infinity if for p ≥ n, the L~p-norm of R?m is finite.As applications, we prove that(M~n, g) is compact if the L~p-norm of R?m is finite and R is positive, and(M~n, g) is scalar flat if(M~n, g) is a complete noncompact manifold with nonnegative scalar curvature and finite L~p-norm of R?m. We prove that(M~n, g) is isometric to a spherical space form if for p ≥n/2, the L~p-norm of R?m is sufficiently small and R is positive.In particular, we prove that(M~n, g) is isometric to a spherical space form if for p ≥ n, R is positive and the L~p-norm of R?m is pinched in [0, C), where C is an explicit positive constant depending only on n, p, R and the Yamabe constant.     

Linear Weingarten Spacelike Submanifolds in Semi-Riemannian Space Form

Mathematical Notes - We consider the linear Weingarten spacelike submanifold immersed in semi-Riemannian space form $$\mathbb {N}^{n+p} _q (c)$$ of constant sectional curvature $$c$$ and index...     

Möbius Homogeneous Hypersurfaces with Three Distinct Principal Curvatures in Sn+1

Let x : M~n→ S~(n+1) be an immersed hypersurface in the(n + 1)-dimensional sphere S~(n+1). If, for any points p, q ∈ Mn, there exists a Mbius transformation φ :S~(n+1)→ S~(n+1) such that φox(Mn~) = x(M~n) and φ ox(p) = x(q), then the hypersurface is called a Mbius homogeneous hypersurface. In this paper, the Mbius homogeneous hypersurfaces with three distinct principal curvatures are classified completely up to a Mbius transformation.     

Lower bounds on Ricci curvature and quantitative behavior of singular sets

Let Y ⁿ denote the Gromov-Hausdorff limit $M^{n}_{i}\stackrel{d_{\mathrm{GH}}}{\longrightarrow} Y^{n}$ of v-noncollapsed Riemannian manifolds with ${\mathrm{Ric}}_{M^{n}_{i}}\geq-(n-1)$ . The singular set $\mathcal {S}\subset Y$ has a stratification $\mathcal {S}^{0}\subset \mathcal {S}^{1}\subset\cdots\subset \mathcal {S}$ , where $y\in \mathcal {S}^{k}$ if no tangent cone at y splits off a factor ? ^k+1 isometrically. Here, we define for all η>0, 0<r≤1, the k-th effective singular stratum $\mathcal {S}^{k}_{\eta,r}$ satisfying $\bigcup_{\eta}\bigcap_{r} \,\mathcal {S}^{k}_{\eta,r}= \mathcal {S}^{k}$ . Sharpening the known Hausdorff dimension bound $\dim\, \mathcal {S}^{k}\leq k$ , we prove that for all y, the volume of the r-tubular neighborhood of $\mathcal {S}^{k}_{\eta,r}$ satisfies ${\mathrm {Vol}}(T_{r}(\mathcal {S}^{k}_{\eta,r})\cap B_{\frac{1}{2}}(y))\leq c(n,{\mathrm {v}},\eta)r^{n-k-\eta}$ . The proof involves a quantitative differentiation argument. This result has applications to Einstein manifolds. Let $\mathcal {B}_{r}$ denote the set of points at which the C ²-harmonic radius is ≤r. If also the $M^{n}_{i}$ are Kähler-Einstein with L ₂ curvature bound, $\| Rm\|_{L_{2}}\leq C$ , then ${\mathrm {Vol}}( \mathcal {B}_{r}\cap B_{\frac{1}{2}}(y))\leq c(n,{\mathrm {v}},C)r^{4}$ for all y. In the Kähler-Einstein case, without assuming any integral curvature bound on the $M^{n}_{i}$ , we obtain a slightly weaker volume bound on $\mathcal {B}_{r}$ which yields an a priori L _p curvature bound for all p<2. The methodology developed in this paper is new and is applicable in many other contexts. These include harmonic maps, minimal hypersurfaces, mean curvature flow and critical sets of solutions to elliptic equations.     

Maximum principles for submanifolds of arbitrary codimension and bounded mean curvature

We construct quadratic forms on which are subharmonic on any n-dimensional minimal submanifold in and, more generally, on submanifolds of bounded mean curvature. This leads to nonexistence results for connected n-dimensional minimal submanifolds in as well as to necessary conditions for the existence of connected submanifolds of bounded mean curvature with arbitrary codimension. Furthermore we discuss a barrier principle for n-dimensional submanifolds in of bounded mean curvature.Received: 11 November 2003, Accepted: 29 January 2004, Published online: 2 April 2004Mathematics Subject Classification (2000): 35J60, 49Q05, 53C42     

Marcinkiewicz integral on weighted Hardy spaces

In this paper we give sufficient conditions to imply the $H^{1}_{w}-L^{1}_{w}$ boundedness of the Marcinkiewicz integral operator $\mu_\Omega$, where w is a Muckenhoupt weight. We also prove that, under the stronger condition $\Omega \in {\rm Lip}_\alpha$, the operator $\mu_\Omega$ is bounded from $H^{p}_{w}$ to $L^{p}_{w}$ for $\max\{n/(n+1/2), n/(n+\alpha)\}$ < p < 1.     

On the Positivity of Scattering Operators for Poincaré-Einstein Manifolds

In this paper, we mainly study the scattering operators for a Poincaré-Einstein manifold $(X^{n+1}, g_+)$, which define the fractional GJMS operators $P_{2\gamma}$ of order $2\gamma$ for $0<\gamma<\frac{n}{2}$ for the conformal infinity $(M, [g])$. We generalise Guillarmou-Qing's positivity results in [8] to the higher order case. Namely, if $(X^{n+1}, g_+)$ $(n\geq 5)$ is a hyperbolic Poincaré-Einstein manifold and there exists a smooth representative $g$ for the conformal infinity such that the scalar curvature $R_g$ is a positive constant and $Q_4$ is semi-positive on $(M, g)$, then $P_{2\gamma}$ is positive for $\gamma\in [1,2]$ and the first real scattering pole is less than $\frac{n}{2}-2$.     

Parallel tangent hyperplanes

Let be a smooth strictly convex closed hypersurface in and let be any oriented smooth connected manifold immersed in Suppose that is a continuous function from to Then there is at least one point such that the hyperplane tangent to at is parallel to the hyperplane tangent to the immersed manifold at the point corresponding to If there did not exist at least two such points, would have to be compact and the Hurewicz homomorphism of into would have to be surjective. If in addition our immersion was an embedding, the Euler characteristic of would have to be equal to For any and any immersed we could always get maps for which the number of points satisfying the conditions of our theorem exactly equaled two. An example can be given in which both and are the unit sphere about the origin in and there is only one such point .