Complete hypersurfaces with constant scalar curvature in spheres

To a given immersion i:Mⁿ? \mathbb Sⁿ⁺¹{i:M^n\to \mathbb S^{n+1}} with constant scalar curvature R, we associate the supremum of the squared norm of the second fundamental form sup |A|². We prove the existence of a constant C _n (R) depending on R and n so that R ≥ 1 and sup |A|² = C _n (R) imply that the hypersurface is a H(r)-torus \mathbb S¹(?{1-r²})×\mathbb S^n-1 (r){\mathbb S^1(\sqrt{1-r^2})\times\mathbb S^{n-1} (r)}. For R > (n − 2)/n we use rotation hypersurfaces to show that for each value C > C _n (R) there is a complete hypersurface in \mathbb Sⁿ⁺¹{\mathbb S^{n+1}} with constant scalar curvature R and sup |A|² = C, answering questions raised by Q. M. Cheng.     

On the weak non-defectivity of veronese embeddings of projective spaces

Fix integers n, x, k such that n≥3, k>0, x≥4, (n, x)≠(3, 4) and k(n+1)<( _n ^n+x ). Here we prove that the order x Veronese embedding ofP ⁿ is not weakly (k−1)-defective, i.e. for a general S⊃P ⁿ such that #(S) = k+1 the projective space | I _2S (x)| of all degree t hypersurfaces ofP ⁿ singular at each point of S has dimension ( _n ^/n+x )−1− k(n+1) (proved by Alexander and Hirschowitz) and a general F∈| I _2S (x)| has an ordinary double point at each P∈ S and Sing (F)=S. The author was partially supported by MIUR and GNSAGA of INdAM (Italy).     

Asymptotic expansions for the distributions of latent roots ofS h S e −1 and of certain test statistics in MANOVA

S _e andS _n are independent central and noncentral Wishart matrices having Wishart distributionsW _p (n _e , Σ) andW _p (n _h , Σ; Ω) respectively. Asymptotic expansions are given for the distributions of latent roots ofS _h S _e ⁻¹ and of certain test statistics in MANOVA under the assumption thatn=n _e +n _h becomes large with a fixed ration _e ∶n _h =e∶h(e+h=1,e>0,h>0) andΩ=O(n).     

Sur les fonctions entièresq-arithmétiques

Letq ɛ Z, |q|>1. In this paper, we study entire functions of a complex variable such thatf(q ^n+m)≡f(qⁿ) (modq ^m-1), ∀n ɛ N andm>0. We prove that iff is of sufficiently small growth, then it is a polynomial.      

Curvature bounds for the spectrum of a compact Riemannian manifold of constant scalar curvature

Let (M, g) be an n-dimensional compact and connected Riemannian manifold of constant scalar curvature. If the sectional curvatures of M are bounded below by a constant α > 0, and the Ricci curvature satisfies Ric < (n − 1)αδ, δ ≥ 1, then it is shown that either M is isometric to the n-sphere Sⁿ(α) or else each nonzero eigenvalue λ of the Laplacian acting on the smooth functions of M satisfies the following:

Inversion and characterization of the hemispherical transform

Explicit inversion formulas are obtained for the hemispherical transform(FΜ)(x) = Μ{y ∃S ⁿ :x. y ≥ 0},x ∃S ⁿ, whereS ⁿ is thendimensional unit sphere in ℝⁿ⁺¹,n ≥ 2, and Μ is a finite Borel measure onS ⁿ. If Μ is absolutely continuous with respect to Lebesgue measuredy onS ⁿ, i.e.,dΜ(y) =f(y)dy, we write(F f)(x) = ∫ _{x.y> 0} f(y)dy and consider the following cases: (a)f ∃C ^∞(Sⁿ); (b)f ∃ L^p(S ⁿ), 1 ≤ p < ∞; and (c)f ∃C(Sⁿ). In the case (a), our inversion formulas involve a certain polynomial of the Laplace-Beltrami operator. In the remaining cases, the relevant wavelet transforms are employed. The range ofF is characterized and the action in the scale of Sobolev spacesL _p ^γ (Sⁿ) is studied. For zonalf ∃ L¹(S ²), the hemispherical transformF f was inverted explicitly by P. Funk (1916); we reproduce his argument in higher dimensions. Partially sponsored by the Edmund Landau Center for Research in Mathematical Analysis, supported by the Minerva Foundation (Germany).     

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).     

Weakly stable constant mean curvature hypersurfaces

Let M^n be an n-dimensional complete noncompact oriented weakly stable constant mean curvature hypersurface in an （n ＋ 1）-dimensional Riemannian manifold N^n＋1 whose （n - 1）th Ricci curvature satisfying Ric^N（n-1） （n - 1）c. Denote by H and φ the mean curvature and the trace-free second fundamental form of M respectively. If ｜φ｜^2 - （n- 2）√n（n- 1）｜H｜｜φ｜＋ n（2n - 1）（H^2＋ c） 〉 0, then M does not admit nonconstant bounded harmonic functions with finite Dirichlet integral. In particular, if N has bounded geometry and c ＋ H^2 〉 0, then M must have only one end.     

Decomposition of the conjugacy representation of the symmetric groups

Consider the two natural representations of the symmetric groupS _n on the group algebra ℂ[S _n ]: the regular representation and the conjugacy representation (acting on the basis by conjugation). Letm(λ) be the multiplicity of the irreducible representationS ^λ in the conjugacy representation and letf ^λ be the multiplicity ofS ^λ in the regular representation. By the character estimates of [R1] and [Wa] we prove

SPACELIKE SUBMANIFOLDS IN THE DE SITTER SPACE Sp^（n＋p）（c）WITH CONSTANT SCALAR CURVATURE

Let M^n be a closed spacelike submanifold isometrically immersed in de Sitter space Sp^(n p)(c)， Denote by R,H and S the normalized scalar curvature,the mean curvature and the square of the length of the second fundamental form of M^n ,respectively. Suppose R is constant and R≤c. The pinching problem on S is studied and a rigidity theorem for M^n immersed in Sp^(n p)(c) with parallel normalized mean curvature vector field is proved. When n≥3, the pinching constant is the best. Thus, the mistake of the paper “Space-like hypersurfaces in de Sitter space with constant scalar curvature”(see Manus Math, 1998,95 :499-505) is corrected. Moreover,the reduction of the codimension when M^n is a complete submanifold in Sp^(n p)(c) with parallel normalized mean curvature vector field is investigated.