Submanifolds with parallel normalized mean curvature vector in a unit sphere

Let M ⁿ be an n-dimensional closed submanifold of a sphere with parallel normalized mean curvature vector. Denote by S and H the squared norm of the second fundamental form and the mean curvature of M ⁿ , respectively. Assume that the fundamental group \({\pi_{1}(M^{n})}\) of M ⁿ is infinite and \({S\, \leqslant\, S(H)=n+\frac{n^{3}H^{2}}{2(n-1)}-\frac{n(n-2)H}{2(n-1)}\sqrt{n^{2}H^{2}+4(n-1)}}\), then S is constant, S = S(H), and M ⁿ is isometric to a Clifford torus \({S^{1}(\sqrt{1-r^{2}})\times S^{n-1}(r)}\) with \({r^{2}\leqslant \frac{n-1}{n}}\).     

CR-submanifolds of complex hyperbolic spaces satisfying a basic equality

The first author introduced a Riemannian invariant denoted by δ and proved in [4] that everyn-dimensional submanifold of the complex hyperbolicm-space ℂH ^m (4c) of constant holomorphic sectional curvature 4c<0 satisfies a basic inequality , whereH ² denotes the squared mean curvature of the submanifold. The main purpose of this paper is to completely classify properCR-submanifolds of complex hyperbolic spaces which satisfy the equality case of this inequality. Dedicated to Leopold Verstraelen on his fiftieth birthday     

On Ricci curvature of isotropic and Lagrangian submanifolds in complex space forms

Let M ⁿ be a Riemannian n-manifold. Denote by S(p) and [`(Ric)](p)\overline {Ric}(p) the Ricci tensor and the maximum Ricci curvature on M <sup>n</sup> at a point p ? M<sup>n</sup>p\in M^n, respectively. First we show that every isotropic submanifold of a complex space form [(M)\tilde]<sup>m</sup>(4 c)\widetilde M^m(4\,c) satisfies S ￡ ((n-1)c+ [(n<sup>2</sup>)/4] H<sup>2</sup>)gS\leq ((n-1)c+ {n^2 \over 4} H^2)g, where H<sup>2</sup> and g are the squared mean curvature function and the metric tensor on M <sup>n</sup>, respectively. The equality case of the above inequality holds identically if and only if either M <sup>n</sup> is totally geodesic submanifold or n = 2 and M <sup>n</sup> is a totally umbilical submanifold. Then we prove that if a Lagrangian submanifold of a complex space form [(M)\tilde]<sup>m</sup>(4 c)\widetilde M^m(4\,c) satisfies [`(Ric)] = (n-1)c+ [(n²)/4] H²\overline {Ric}= (n-1)c+ {n^2 \over 4} H^2 identically, then it is a minimal submanifold. Finally, we describe the geometry of Lagrangian submanifolds which satisfy the equality under the condition that the dimension of the kernel of second fundamental form is constant.     

Complete Hypersurfaces with Constant Mean Curvature in a Unit Sphere

By investigating hypersurfaces M ⁿ in the unit sphere S ⁿ⁺¹(1) with constant mean curvature and with two distinct principal curvatures, we give a characterization of the torus S ¹(a) × , where . We extend recent results of Hasanis et al. [5] and Otsuki [10].     

L 2 harmonic forms and stability of hypersurfaces with constant mean curvature

We prove that a complete noncompact oriented strongly stable hypersurfaceM ⁿ with cmc (constant mean curvature)H in a complete oriented manifoldN ⁿ⁺¹ with bi-Ricci curvature, satisfying alongM, admits no nontrivialL ² harmonic 1-forms. This implies ifM ⁿ (2n4) is a complete noncompact strongly stable hypersurface in hyperbolic spaceH ⁿ⁺¹(–1) with cmc , there exist no nontrivialL ² harmonic 1-forms onM. We also classify complete oriented strongly stable surfaces with cmcH in a complete oriented manifoldN ³ with scalar curvature satisfying .     

Factorial series connected with the Lambert function,and a problem posed by Ramanujan

Ramanujan’s sequence {y _n } _n=0^∞ defined by is expanded in factorial series derived from a series representing the Lambert W function. As a corollary, it is shown that the sequence {y _n } is completely monotonic.      

A basic inequality and new characterization of Whitney spheres in a complex space form

LetN ⁿ (4c) be ann-dimensional complex space form of constant holomorphic sectional curvature 4c and letx:M ⁿ →N ⁿ (4c) be ann-dimensional Lagrangian submanifold inN ⁿ (4c). We prove that the following inequality always hold onM ⁿ: whereh is the second fundamental form andH is the mean curvature of the submanifold. We classify all submanifolds which at every point realize the equality in the above inequality. As a direct consequence of our Theorem, we give, a new characterization of theWhitney spheres in a complex space form. Partially supported by a research fellowship of the Alexander von Humboldt Stiftung.     

On the sum Σ k≡r(modm) ( k n ) and related congruencesand related congruences

In this paper we study [ _r ⁿ ] _m =Σ _k≡r(modm) ( _k ⁿ ) wherem>0,n≥0 andr are integers. We show that [ _r ⁿ ] _m (m>2) can be expressed in terms of some linearly recurrent sequences with orders not exceeding ϕ(m)/2. In particular, we determine [ _r ⁿ ] ₁₂ explicitly in terms of first order and second order recurrences. It follows that for any primep>3 we have and . The research is supported by the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE, and the National Natural Science Foundation of P. R. China.     

On hypothesis K for biquadrates

It is proved that the number of ways of expressing a large positive integern as the sum of four biquadrates is

C-Totally real pseudo-parallel submanifolds of Sasakian space forms

Let be (2n + 1)-dimensional Sasakian space form of constant ϕ-sectional curvature (c) and M ⁿ be an ⁿ -dimensional C-totally real, minimal submanifold of . We prove that if M ⁿ is pseudo-parallel and , then M ⁿ is totally geodesic.     

Inequalities for a polynomial and its derivative

For an arbitrary entire functionf and anyr>0, letM(f,r):=max_|z|=r |f(z)|. It is known that ifp is a polynomial of degreen having no zeros in the open unit disc, andm:=min_|z |=1|p(z)|, then

Precise asymptotics in the Baum-Katz and davis law of large numbers for positively associated sequences

§ 1　 Introduction and resultsL et { X,Xi;i≥ 1} be a sequence of i.i.d.random variables,and set Sn= ni=1 Xi,n≥1.Hsu and Robbins[1 ] introduced the conceptof complete convergence.They together withErdos[2 ] proved n≥ 1 P(|Sn|≥εn) <∞ ,ε>0 (1)if and only if EX=0 and EX2 <∞ .L ater,Spitzer[3] proved n≥ 11n P(|Sn|≥εn) <∞ ,ε>0if and only if EX =0 and E|X|<∞ .More generally,it was shown by Baum and Katz[4 ]that,for 0 0 (…     

Remarks on Spectral Radius and Laplacian Eigenvalues of a Graph

Let G be a graph with n vertices, m edges and a vertex degree sequence (d ₁, d ₂,..., d _n ), where d ₁ ≥ d ₂ ≥ ... ≥ d _n . The spectral radius and the largest Laplacian eigenvalue are denoted by ϱ(G) and μ(G), respectively. We determine the graphs with

Hybrid triple systems and cubic feedback sets

Ac-hybrid triple system of orderv is a decomposition of the completev-vertex digraph intoc cyclic tournaments of order 3 and transitive tournaments of order 3. Hybrid triple systems generalize directed triple systems (c = 0) and Mendelsohn triple systems (c = v(v – 1)/3); omitting directions yields an underlying twofold triple system. The spectrum ofv andc for which ac-hybrid triple system of orderv exists is completely determined in this paper. Using (cubic) block intersection graphs, we then show that every twofold triple system of order underlies ac-hybrid triple system with . Examples are constructed for all sufficiently largev, for which this maximum is at most . The lower bound here is proved by establishing bounds onF _i (n, r), the size of minimum cardinality vertex feedback sets inn-vertexi-connected cubic multigraphs havingr repeated edges. We establish that , 8$$ " align="middle" border="0"> . These bounds are all tight, and the latter is used to derive the lower bound in the design theoretic problem.     

On the complete monotonicity of a Ramanujan sequence connected with e n

We show that the Ramanujan sequence (θ _n ) _n≥0 defined as the solution to the equation

Asymptotic properties for the loglog laws under positive association

Let {X _n : n ?? 1} be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set $S_n = \sum\limits_{k = 1}^n {X_k }$ , $Mn = \mathop {\max }\limits_{k \leqslant n} \left| {S_k } \right|$ , n ?? 1. Suppose that $0 < \sigma ^2 = EX_1^2 + 2\sum\limits_{k = 2}^\infty {EX_1 X_k < \infty }$ . In this paper, we prove that if E|X ₁|^2+?? < for some ?? ?? (0, 1], and $\sum\limits_{j = n + 1}^\infty {Cov\left( {X_1 ,X_j } \right) = O\left( {n^{ - \alpha } } \right)}$ for some ?? > 1, then for any b > ?1/2 $$\mathop {\lim }\limits_{\varepsilon \searrow 0} \varepsilon ^{2b + 1} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^{b - 1/2} }} {{n^{3/2} \log n}}} E\left\{ {M_n - \sigma \varepsilon \sqrt {2n\log \log n} } \right\}_ + = \frac{{2^{ - 1/2 - b} E\left| N \right|^{2(b + 1)} }} {{(b + 1)(2b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2(b + 1)} }}}$$ and $$\mathop {\lim }\limits_{\varepsilon \nearrow \infty } \varepsilon ^{ - 2(b + 1)} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^b }} {{n^{3/2} \log n}}E\left\{ {\sigma \varepsilon \sqrt {\frac{{\pi ^2 n}} {{8\log \log n}}} - M_n } \right\}} _ + = \frac{{\Gamma (b + 1/2)}} {{\sqrt 2 (b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2b + 2} }}} ,$$ where x ₊ = max{x, 0}, N is a standard normal random variable, and ??(·) is a Gamma function.     

How often is a polygon bounded by three sides?

LetL _n be the set of lines (no two parallel) determining ann-sided bounded faceF in the Euclidean plane. We show that the number,f(L _n), of triples fromL _n that determine a triangle containingF satisfies and these bounds are best. This result is generalized tod-dimensional Euclidean space (without the claim that the upper bound is attainable).     

Geometric, topological and differentiable rigidity of submanifolds in space forms

Let M be an n-dimensional submanifold in the simply connected space form F ^n+p (c) with c + H ² > 0, where H is the mean curvature of M. We verify that if M ⁿ (n ≥ 3) is an oriented compact submanifold with parallel mean curvature and its Ricci curvature satisfies Ric _M ≥ (n ? 2)(c + H ²), then M is either a totally umbilic sphere, a Clifford hypersurface in an (n + 1)-sphere with n = even, or ${\mathbb{C}P^{2} \left(\frac{4}{3}(c + H^{2})\right) {\rm in} S^{7} \left(\frac{1}{\sqrt{c + H^{2}}}\right)}$ C P 2 4 3 ( c + H 2 ) in S 7 1 c + H 2 . In particular, if Ric _M > (n ? 2)(c + H ²), then M is a totally umbilic sphere. We then prove that if M ⁿ (n ≥ 4) is a compact submanifold in F ^n+p (c) with c ≥ 0, and if Ric _M > (n ? 2)(c + H ²), then M is homeomorphic to a sphere. It should be emphasized that our pinching conditions above are sharp. Finally, we obtain a differentiable sphere theorem for submanifolds with positive Ricci curvature.     

Ricci curvature and quasiconformal deformations of a Riemannian manifold

The theory of quasiconformal deformations of a Riemannian manifold (M, g) of dimensionn leads in a natural way to the AhlforsS operator being the symmetric and trace free part of the Levi-Civita connection Δ on 1-forms, and to the Ahlfors Laplacian , whereR is the Ricci action. It is well known that there are no conformal deformations on compact Riemannian manifoldsM with negative Ricci curvature. The question arises, how close to being conformal a deformation on suchM can be, i.e. the question on the minimal constant of quasiconformality. Using spectral properties ofL, we derive several lower bounds for the constant of quasiconformality for the normalized deformations of compact manifolds with the positive definite, negative definite, or vanishing Ricci tensorR. As a result, we also obtain that there are no conformal deformations if the Ricci curvature is positive definite but small enough (Corollary 5.17).     

The number of partial orders of fixed width

We investigate the approximate number of n-element partial orders of width k, for each fixed k. We show that the number of width 2 partial orders with vertex set {1, 2, ..., n} is