THE FIRST EIGENVALUE OF AN IRREDUCIBLE HOMOGENEOUS MANIFOLD

Let M be an n-dimensional compact minimal submanifold in the unit sphere. It is shown that the diameter and volume of M satisfyd≥π/2+C(n)d~n/(d~n+V).An application is that if M is an n-dimensional compact irreducible homogeneous manifold, the first eigenvalue λ_1 of M satisfiesλ_1≥n/d~2(π/2+C(n)d~n/(d~n+V))~2.In the above two cases, C(n)'s are the same constants depending only on n.     

Liouville Type Theorem About p-Harmonic Function and p-Harmonic Map with Finite L-Energy

This paper deals with the p-harmonic function on a complete non-compact submanifold M isometrically immersed in an (n + k)-dimensional complete Riemannian manifold (M) of non-negative (n-1)-th Ricci curvature.The Liouville type theorem about the p-harmonic map with finite Lq-energy from complete submanifold in a partially nonnegatively curved manifold to non-positively curved manifold is also obtained.     

Hamiltonian Systems with Positive Topological Entropy and Conjugate Points

In this article, we consider the geodesic flows induced by the natural Hamiltonian systems $H(x,p)=\frac{1}{2}g^{ij}(x) p_{i}p_{j} + V(x) $ defined on a smooth Riemannian manifold$(M = \mathbb{S}^{1} \times N, g)$, where $\mathbb {S}^{1}$ is the one dimensional torus, N is a compact manifold, g is the Riemannian metric on M and V is a potential function satisfying $V \leq 0$. We prove that under suitable conditions, if the fundamental group $\pi_{1}(N)$ has sub-exponential growth rate, then the Riemannian manifold M with the Jacobi metric $(h-V)g$, i.e., $(M, (h-V)g)$, is a manifold with conjugate points for all h with $0 < h <\delta$, where $\delta$ is a small number.     

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.     

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.     

A spinorial characterization of hyperspheres

Let M be a compact orientable n-dimensional hypersurface, with nowhere vanishing mean curvature H, immersed in a Riemannian spin manifold ${\overline{M}}$ admitting a non trivial parallel spinor field. Then the first eigenvalue ${\lambda_1(D_{M}^{H})}$ (with the lowest absolute value) of the Dirac operator ${D_{M}^{H}}$ corresponding to the conformal metric ${\langle\;,\;\rangle^{H}=H^{2}\,\langle\;,\;\rangle}$ , where ${\langle\;,\;\rangle}$ is the induced metric on M, satisfies ${\left|\lambda_1(D_{M}^{H})\right|\le \frac{n}{2}}$ . By applying the Bourguignon-Gauduchon first variational formula, we obtain a necessary condition for ${\left|\lambda_1(D_{M}^{H})\right|=\frac{n}{2}}$ . As a consequence, we prove that round hyperspheres are the only hypersurfaces of the Euclidean space satisfying the equality in the Bär inequality $$\lambda_1(D_{M})^{2}\le \frac{n^{2}}{4{vol}(M)}\int_{M} H^{2}\, dV,$$ where D _M stands now for the Dirac operator of the induced metric.     

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.     

WEAK TYPE(1,1)BOUNDEDNESS OF RIESZ TRANSFORM ON POSITIVELY CURVED MANIFOLDS

For complete Riemannian manifold M,it is proved that▽(-△)~(-1/2) is boundedfrom L~2(M)to weak-L~1(M)if Ric(M)≥0.     

AN INVERSE EIGENVALUE PROBLEM FOR JACOBI MATRICES

Let T1,n be an n x n unreduced symmetric tridiagonal matrix with eigenvaluesand is an (n - 1) x (n - 1) submatrix by deleting the kth row and kth column, k = 1, 2,be the eigenvalues of T1,k andbe the eigenvalues of Tk+1,nA new inverse eigenvalues problem has put forward as follows: How do we construct anunreduced symmetric tridiagonal matrix T1,n, if we only know the spectral data: theeigenvalues of T1,n, the eigenvalues of Ti,k-1 and the eigenvalues of Tk+1,n?Namely if we only know the data: A1, A2, An,how do we find the matrix T1,n? A necessary and sufficient condition and an algorithm ofsolving such problem, are given in this paper.     

ON THE HARNACK INEQUALITY FOR HARMONIC FUNCTIONS ON COMPLETE RIEMANNIAN MAINFOLDS

First it is shown that on the complete Riemannian manifold with nonnegative Ricci curvature $\overline M$ the Sobolev type inequality $\[||\nabla u|{|_2} \geqslant {C_{n,\alpha }}||u|{|_{2\alpha }}(\alpha \geqslant 1)\]$, for all $u \in H^2_1(\overline M)$ holds if and only if $V_x(r)=Vol(B_x(r))\geq C_nr^n$ and $\alpha=\frac{n}{n-2}$. Let M be a complete Riemannian manifolds which is uniformly equivalent to $\overline M$, and assume that $V_x(r)\geq C_nr^n$ on $\overline M$. Then it is prioved that the John-Nirenberg inequality, holds on M. Finally, based on the Sobolev inequality and John-Nirenberg inequality, the Harnack inequality for harmonic functions on M is obtained by the method of Moser, arid consequently some Liouville theorems for harmonic functions and harmonic maps on M are proved.     

$L^p$ Harmonic $k$-Forms on Complete Noncompact Hypersurfaces in $\mathbb{S}^{n+1}$ with Finite Total Curvature

In general, the space of $L^p$ harmonic forms $\mathcal{H}^k(L^p(M))$ and reduced $L^p$ cohomology $H^k(L^p(M))$ might be not isomorphic on a complete Riemannian manifold $M$, except for $p=2$. Nevertheless, one can consider whether $\mathrm{dim}\mathcal{H}^k(L^p(M))<+\infty$ are equivalent to $\mathrm{dim}H^k(L^p(M))<+\infty$. In order to study such kind of problems, this paper obtains that dimension of space of $L^p$ harmonic forms on a hypersurface in unit sphere with finite total curvature is finite, which is also a generalization of the previous work by Zhu. The next step will be the investigation of dimension of the reduced $L^p$ cohomology on such hypersurfaces.     

具有几乎最大深度的滤链的Hilbert 系数

设${\cal F}$是Cohen-Macaulay R-模的一个Hilbert滤链. 当$G({\cal F},M)$和$F_K({\cal F},M)$有几乎最大深度时,我们证明长度$\lambda(KI_nM/JI_{n-1}M)$和约化数$r^K_J({\cal F},M)$与$J$无关,并且给出了第一和第二个Hilbert 系数的下界.     

THE EXISTENCE OF CLOSE GEODESICS ON A COMPLETE RIEM ANNIAN MANIFOLD

This paper studios the existence of closed geodesics in the homotopy class of a given closed curve. Let M be a complete Riemannian manifold without boundary, f∈C~1(S~1, M). Look at S~1 as [0, 2π]/{0, 2π}. The following results are proved:A. The initial value problem of heat equation _if_t=τ(f_i), f_0=f always admits a global solution.B. (Existence of closed geodesics). If there exists a compact set KM such that f(S~1)∩K≠φ andE(f)≤(1/π)l(K)~2,then there exists a closed geodesic homotopic to f. If then the closed geodesic is minimal.C. Some estimates about injective radius are obtained.Some example is found showing that the inequalities in B are sharp.     

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   

The stability of Hausdorff dimension for the level sets under the perturbation of conformal repellers

Let $M$ be a $C^\infty$ compact Riemann manifold. $f:M\to M$ is a $C^1$ map and $\Lambda_f \subset M$ is a conformal repeller of $f$. Suppose $\varphi:M\to\mathbb{R}$ is a continuous function and let $f_k$ be nonconformal perturbation of the map $f$. We consider the stability of Hausdorff dimension of level sets for Birkhorff average of potential function $\varphi$ with respect to $f_k$ and $f$.     

Geometry of the Second-Order Tangent Bundles of Riemannian Manifolds

Let (M,g) be an n-dimensional Riemannian manifold and T2M be its secondorder tangent bundle equipped with a lift metric (g).In this paper,first,the authors construct some Riemannian almost product structures on (T2M,(g)) and present some results concerning these structures.Then,they investigate the curvature properties of (T2M,(g)).Finally,they study the properties of two metric connections with nonvanishing torsion on (T2 M,(g)):The H-lift of the Levi-Civita connection of g to T2 M,and the product conjugate connection defined by the Levi-Civita connection of (g) and an almost product structure.     

On the General Algebraic Inverse Eigenvalue Problems

A number of new results on sufficient conditions for the solvability and numerical algorithms of the following general algebraic inverse eigenvalue problem are obtained: Given $n+1$ real $n\times n$ matrices $A=(a_{ij}),A_k=(a_{ij}^{(k)})(k=1,2,\cdots,n)$ and $n$ distinct real numbers $\lambda_1,\lambda_2,\cdots,\lambda_n,$ find $n$ real number $c_1,c_2,\cdots,c_n$ such that the matrix $A(c)=A+\sum\limits_{k=1}^{n}c_k A_k$ has eigenvalues $\lambda_1,\lambda_2,\cdots,\lambda_n.$     

一类奇型Sturm-Liouville算子的逆问题

研究了奇型Sturm-Liouville算子的逆问题.对于固定的n∈N,证明了Sturm-Liouville问题(1.3)-(1.5)的第n个特征值λ_n(q,H)关于H是严格单调增加的,及一组不同边界条件下的第n个特征值的谱集合{λ_n(q,H_k)}_(k=1)~(+∞)能够唯一确定(0,πr)上的势函数q(x).     

$L^{p}_{[0,1]}$空间M\"{u}ntz有理逼近的Jackson型估计 (英)

设$\Lambda=\{\lambda_{n}\}_{n=1}^{\infty}$为正的实数数列, 且当$n\rightarrow\infty$时, 有$\lambda_{n}\searrow 0$.本文给出了当 $\lambda_{n}\leq Mn^{-\frac{1}{2}},\;n=1,2, \cdots ,$(其中$M>0$为一正常数)时M\"{u}ntz系统$\{x^{\lambda_n}\}$的有理函数在$ L_{[0,1]} ^{p}$空间的逼近速度,主要结论为$R_{n} (f, \Lambda )_{L^{p}}\leq C_M \omega (f, n^{-\frac{1}{2}})_{L^{p}},\;1 \leq p \leq \infty.$     

Extremal Functions for Adams Inequalities in Dimension Four

Let $\Omega\subset \mathbb{R}^4$ be a smooth bounded domain, $W_0^{2,2}(\Omega)$ be the usual Sobolev space. For any positive integer $\ell$, $\lambda_{\ell}(\Omega)$ is the $\ell$-th eigenvalue of the bi-Laplacian operator. Define $E_{\ell}=E_{\lambda_1(\Omega)}\oplus E_{\lambda_2(\Omega)}\oplus\cdots\oplus E_{\lambda_{\ell}(\Omega)}$, where $E_{\lambda_i(\Omega)}$ is eigenfunction space associated with $\lambda_i(\Omega)$. $E^{\bot}_{\ell}$ denotes the orthogonal complement of $E_\ell$ in $W_0^{2,2}(\Omega)$. For $0\leq\alpha<\lambda_{\ell+1}(\Omega)$, we define a norm by $\|u\|_{2,\alpha}^{2}=\|\Delta u\|^2_2-\alpha \|u\|^2_2$ for $u\in E^\bot_{\ell}$. In this paper, using the blow-up analysis, we prove the following Adams inequalities$$\sup_{u\in E_{\ell}^{\bot},\,\| u\|_{2,\alpha}\leq 1}\int_{\Omega}e^{32\pi^2u^2}{\rm d}x<+\infty;$$moreover, the above supremum can be attained by a function $u_0\in E_{\ell}^{\bot}\cap C^4(\overline{\Omega})$ with $\|u_0\|_{2,\alpha}=1$. This result extends that of Yang (J. Differential Equations, 2015), and complements that of Lu and Yang (Adv. Math. 2009) and Nguyen (arXiv: 1701.08249, 2017).