Generalized Ejiri's Rigidity Theorem for Submanifolds in Pinched Manifolds(In memory of Professor Chaohao Gu on his 90th birthday)

Let M~n(n ≥ 4) 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 δ(n, p) ∈(0, 1) such that if the sectional curvature of N satisfies■ , 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■ , a Clifford hypersurface S~m■ in the totally umbilic sphere ■, or■ . This is a generalization of Ejiri's rigidity theorem.     

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.     

Rigidity of closed submanifolds in a locally symmetric Riemannian manifold

Let M~n(n ≥ 4) be an oriented closed submanifold with parallel mean curvature in an(n + p)-dimensional locally symmetric Riemannian manifold N~(n+p). We prove that if the sectional curvature of N is positively pinched in [δ, 1], and the Ricci curvature of M satisfies a pinching condition, then M is either a totally umbilical submanifold, or δ = 1, and N is of constant curvature. This result generalizes the geometric rigidity theorem due to Xu and Gu[15].     

A NOTE ON COMPLETE MANIFOLDS WITH FINITE VOLUME

In this article, we concern on complete manifolds with finite volume. We prove that under some assumptions about scalar curvature and the Yamabe constant, the manifolds must be compact, and we also give the diameter estimates in terms of the scalar curvature and the Yamabe constant.     

关于容许全脐超曲面正交族的Einstein流形

The aim of the present paper is to study globally the Riemannian manifold admitting two or more mutually orthogonal families of totally umbilical hypersurfaccs of which each is Einsteinian. This paper consists of four parts: (i) to establish anew the canonical form of the metric of (M,g) admitting p (p≥2) families of mutually orthogonal totally umbilical hypcrsurf aces from the standpoint of global differential geometry; (ii) to prove in a n-dimensional (n>2) Einsteinian manifold E_n of nonvanishing scalar curvature there doesn't exist one family of compact totally geodesic Einsteinian hypersurfaces (Theorem 1);(iii) to prove in a n-dimensional (n≥5) Einsteinian manifold E_n of nonnegative scalar curvature R there don't exist two orthogonal families of totally umbilical but not geodesic complete Einsteinian hypersurfaces (Theorem Ⅱ);(iv) to show that a n-dimensional (n≥5) Riemannian manifold of negative constant scalar curvature R.     

On complete submanifolds with parallel mean curvature in negative pinched manifolds

A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian (n p)-dimensional manifold Nn p with negative sectional curvature is proved. For given positive integers n(≥ 2), p and for a constant H satisfying H ＞ 1 there exists a negative number τ(n,p, H) ∈ (-1, 0) with the property that if the sectional curvature of N is pinched in [-1, τ(n,p, H)], and if the squared length of the second fundamental form is in a certain interval, then Nn p is isometric to the hyperbolic space Hn p(-1). As a consequence, this submanifold M is congruent to Sn(1/ H2-1) or theVeronese surface in S4(1/√H2-1).     

On Einsteinian Manifolds Admitting Orthogonal Families of Totally Umbilical Hypersurfaces

The aim of the present paper is to study globally the Riemannian manifold admitting two or more mutually orthogonal families of totally umbilieal hypersurfaces of which each is Einsteinian. This paper consists of four parts: (i) to establish anew the canonical form of the metric of (M,g)admitting p (p≥2) families of mutually orthogonal totally umbilical hypersurfaces from the standpoint of global differential geometry; (ii) to prove in a n-dimensional (n>2) Einsteinian manifold E_n of nonvanishing scalar curvature there doesn't exist one family of compact totally geodesic Einsteinian hypersurfaces (Theorem I); (iii) to prove in a n-dimensional (n≥5) Einsteinian manifold E, of nonnegative scalar curvature there don't exist two orthogonal families of totally umbilical but not geodesic complete Einsteinian hypersurfaces (Theorem II); (iv) to show that a n-dimensional (n≥5) Riemannian manifold of negative constant scalar curvature admitting p (p≥3) mutually orthogonal families of compact, totally umbili     

具有非负Ricci曲率和大体积增长的开流形

In this paper,we prove that a complete n-dimensional Riemannian manifold with n0nnegative kth-Ricci curvature,large volume growth has finite topological type provided that lim{((vol[B(p,r))]/(ω_nr~n)-αM)r(k(n-1))/(k 1)(1-α/2)}＜=εfor some constantε＞0.We also prove that a complete Riemannian manifold with nonnegative kth-Ricci curvature and under some pinching conditions is diffeomorphic to R~n.     

ON THE SECTIONAL CURVATURE OF A RIEMANNIAN MANIFOLD

In this paper the author establishes the following1.If M~n(n≥3)is a connected Riemannian manifold,then the sectional curvatureK(p),where p is any plane in T~x(M),is a function of at most n(n-1)/2 variables.Moreprecisely,K(p)depends on at most n(n-1)/2 parameters of group SO(n).2.Lot M~n(n≥3)be a connected Riemannian manifold.If there exists a point x ∈ Msuch that the sectional curvature K(p)is independent of the plane p∈T_x(M),then M is aspace of constant curvature.This latter improves a well-known theorem of F.Schur.     

小Excess与具负下界Ricci曲率开流形的拓扑

In this paper,we study the relation between the excess of open manifolds and their topology by using the methods of comparison geometry.We prove that a complete open Riemmannian manifold with Ricci curvature negatively lower bounded is of finite topological type provided that the conjugate radius is bounded from below by a positive constant and its Excess is bounded by some function of its conjugate radius,which improves some results in [4].     

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.     

关于一类弱-Berwald的$(\alpha,\beta)$-度量

In this paper, we study an important class of （α,β）-metrics in the form F = （α＋β）^m＋1/α^m on an n-dimensional manifold and get the conditions for such metrics to be weakly- Berwald metrics, where α = √aij（x）y^iy^j is a Riemannian metric and β = bi（x）y^i is a 1-form and m is a real number with m ≠ -1,0,-1/n. Furthermore, we also prove that this kind of （α,β）-metrics is of isotropic mean Berwald curvature if and only if it is of isotropic S-curvature. In this case, S-curvature vanishes and the metric is weakly-Berwald metric.     

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.     

置换多项式$x^{1+\frac{q-1}{m}}+ax$

设$m$为正整数, $F_{q^r}$是特征为$p$的有限域. 本文证明了如果$p>m^2-m$且$q\equiv 1\pmod{m}$, 则多项式$x^{1+\frac{q-1}{m}}+ax~(a\neq0)$不是$F_{q^r}~(r\geq2)$上的置换多项式. 本文还证明了$q\equiv 1\pmod{7}$且$p\neq 2, 3$时, $x^{1+\frac{q-1}{7}}+ax~(a\neq0)$不是$F_{q^r}~(r\geq2)$上的置换多项式     

两类具有标量旗曲率的弱Berwald度量

In this paper, we study the （α,β）-metrics of scalar flag curvature in the form of F = α ＋ εβ ＋ κβ^2/α （ε and k ≠ 0 are constants） and F = α^2/α-β. We prove that these two kinds of metrics are weak Berwaldian if and only if they are Berwaldian and their flag curvatures vanish. In this case, the metrics are locally Minkowskian.     

一类上三角矩阵环$W_{n}(p, q)$的斜Armendariz性质

设α是环R的一个自同态,称环R是α-斜Armendariz环,如果在R[x;α]中,(∑_(i=0)~ma_ix~i)(∑_(j=0)~nb_jx~j)=0,那么a_ia~i(b_j)=0,其中0≤i≤m,0≤j≤n.设R是α-rigid环,则R上的上三角矩阵环的子环W_n(p,q)是α~—-斜Armendariz环.     

对流扩散方程的解

关注如下的对流扩散方程 $$ u_{t}=\text{div}(|\nabla u^{m}|^{p-2}\nabla u^{m})+\sum_{i=1}^{N}\frac{\partial b_{i}(u^{m})}{\partial x_{i}} $$ 的初边值问题. 若 $p>1+\frac{1}{m}$, 通过考虑正则化问题的解 $u_{k}$, 利用 Moser 迭代技巧, 得到了$u_{k}$ 的 $L^{\infty}$ 模与 梯度 $\nabla u_{k}$ 的 $L^{p}$ 模的局部有界性. 利用紧致性定理, 得到了对流扩散方程本身解的存在性. 若 $p<1+\frac{1}{m},\ p>2$ 或者 $p=1+\frac{1}{m}$, 利用类似的方法可以得到解的存在性. 证明了解的唯一性, 同时讨论了正性和熄灭性等解的性质.     

广义复合二项风险模型的若干大偏差结果

This paper is a further investigation of large deviation for partial and random sums of random variables, where {Xn,n ≥ 1} is non-negative independent identically distributed random variables with a common heavy-tailed distribution function F on the real line R and finite mean μ∈ R. {N（n）,n ≥ 0} is a binomial process with a parameter p ∈ （0,1） and independent of {Xn,n ≥ 1}; {M（n）,n ≥ 0} is a Poisson process with intensity λ 〉 0, Sn = ΣNn i=1 Xi-cM（n）. Suppose F ∈ C, we futher extend and improve some large deviation results. These results can apply to certain problems in insurance and finance.