An application of maximum principle to space-like hypersurfaces with constant mean curvature in anti-de Sitter space

In this paper, we study complete hypersurfaces with constant mean curvature in anti-de Sitter space ${H^{n+1}_1(-1)}$ . we prove that if a complete space-like hypersurface with constant mean curvature ${x:\mathbf M\rightarrow H^{n+1}_1(-1) }$ has two distinct principal curvatures ??, ??, and inf|?? ? ??|?>?0, then x is the standard embedding ${ H^{m} (-\frac{1}{r^2})\times H^{n-m} ( -\frac{1}{1 - r^2} )}$ in anti-de Sitter space ${ H^{n+1}_1 (-1) }$ .     

On complete noncompact submanifolds with constant mean curvature and finite total curvature in Euclidean spaces

In this paper, we consider a complete noncompact n-submanifold M with parallel mean curvature vector h in an Euclidean space. If M has finite total curvature, we prove that M must be minimal, so that M is an affine n-plane if it is strongly stable. This is a generalization of the result on strongly stable complete hypersurfaces with constant mean curvature in Received: 30 June 2005     

On a conjectured inequality of Gautschi and Leopardi for Jacobi polynomials

Motivated by work on positive cubature formulae over the spherical surface, Gautschi and Leopardi conjectured that the inequality holds for α,β > − 1 and n ≥ 1, θ ∈ (0, π), where are the Jacobi polynomials of degree n and parameters (α, β). We settle this conjecture in the special cases where .      

Invariant submanifolds of Sasakian space forms

In the present study, we consider isometric immersions ${f : M \rightarrow \tilde{M}(c)}$ of (2n + 1)-dimensional invariant submanifold M ²ⁿ⁺¹ of (2m + 1) dimensional Sasakian space form ${\tilde{M}^{2m+1}}$ of constant ${ \varphi}$ -sectional curvature c. We have shown that if f satisfies the curvature condition ${\overset{\_}{R}(X, Y) \cdot \sigma =Q(g, \sigma)}$ then either M ²ⁿ⁺¹ is totally geodesic, or ${||\sigma||^{2}=\frac{1}{3}(2c+n(c+1)),}$ or ${||\sigma||^{2}(x) > \frac{1}{3}(2c+n(c+1)}$ at some point x of M ²ⁿ⁺¹. We also prove that ${\overset{\_ }{R}(X, Y)\cdot \sigma = \frac{1}{2n}Q(S, \sigma)}$ then either M ²ⁿ⁺¹ is totally geodesic, or ${||\sigma||^{2}=-\frac{2}{3}(\frac{1}{2n}\tau -\frac{1}{2}(n+2)(c+3)+3)}$ , or ${||\sigma||^{2}(x) > -\frac{2}{3}(\frac{1}{2n} \tau (x)-\frac{1}{2} (n+2)(c+3)+3)}$ at some point x of M ²ⁿ⁺¹.     

Remarks on noncompact steady gradient Ricci solitons

Assume Mⁿ{\mathcal{M}^n} is a complete noncompact steady gradient Ricci soliton with positive Ricci curvature. First, by deriving a useful formula we characterize the condition of the scalar curvature and the potential function having a same level surface. Then, we assume the dimension n = 3 and characterize the rotational symmetry geometrically. Finally, for all dimensions n ≥ 3, we prove a dimension reduction result at spatial infinity under additional assumptions that Mⁿ{\mathcal M^n} is a κ-solution and the scalar curvature is O(\frac1r),{O\left(\frac{1}{r}\right),} where r is the distance function.     

Constant mean curvature hypersurfaces with two principal curvatures in a sphere

In this paper we consider a compact oriented hypersurface M ⁿ with constant mean curvature H and two distinct principal curvatures λ and μ with multiplicities (n − m) and m, respectively, immersed in the unit sphere S ⁿ⁺¹. Denote by the trace free part of the second fundamental form of M ⁿ , and Φ be the square of the length of . We obtain two integral formulas by using Φ and the polynomial . Assume that B _H,m is the square of the positive root of P _H,m (x) = 0. We show that if M ⁿ is a compact oriented hypersurface immersed in the sphere S ⁿ⁺¹ with constant mean curvatures H having two distinct principal curvatures λ and μ then either or . In particular, M ⁿ is the hypersurface .      

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-harmonic forms and stable hypersurfaces in space forms

We prove that a complete noncompact orientable stable minimal hypersurface in \mathbbSⁿ⁺¹{\mathbb{S}^{n+1}} (n ≤ 4) admits no nontrivial L ²-harmonic forms. We also obtain that a complete noncompact strongly stable hypersurface with constant mean curvature in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} or \mathbbSⁿ⁺¹{\mathbb{S}^{n+1}} (n ≤ 4) admits no nontrivial L ²-harmonic forms. These results are generalized versions of Tanno’s result on stable minimal hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}}.     

Stable minimal hypersurfaces in a Riemannian manifold with pinched negative sectional curvature

We give an estimate of the smallest spectral value of the Laplace operator on a complete noncompact stable minimal hypersurface M in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space, we prove that if a complete minimal hypersurface M has sufficiently small total scalar curvature then M has only one end. We also obtain a vanishing theorem for L ² harmonic 1-forms on minimal hypersurfaces in a Riemannian manifold with sectional curvature bounded below by a negative constant. Moreover, we provide sufficient conditions for a minimal hypersurface in a Riemannian manifold with nonpositive sectional curvature to be stable.     

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.     

On the structure of complete hypersurfaces in a Riemannian manifold of nonnegative curvature and L 2 harmonic forms

Let M ⁿ be a complete oriented noncompact hypersurface in a complete Riemannian manifold N ⁿ⁺¹ of nonnegative sectional curvature with ${2 \leq n \leq 5}$ . We prove that if M satisfies a stability condition, then there are no non-trivial L ² harmonic one-forms on M. This result is a generalization of a well-known fact in the case when M is a stable minimally immersed hypersurface. As a consequence, we show that if the mean curvature of M is constant, then either M must have only one end or M splits into a product of ${\mathbb{R}}$ and a compact manifold with nonnegative sectional curvature. In case ${n \geq 5}$ , we also show that the same result holds if the absolute value of the mean curvature is less than or equal to the ratio of the norm of the second fundamental form to the dimension of a hypersurface.     

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 .     

On the Shape of the Ground State Eigenfunction for Stable Processes

We prove that the ground-state eigenfunction for symmetric stable processes of order α∈(0,2) killed upon leaving the interval (?1,1) is concave on $(-\frac{1}{2},\frac{1}{2})We prove that the ground-state eigenfunction for symmetric stable processes of order α∈(0,2) killed upon leaving the interval (−1,1) is concave on . We call this property “mid-concavity”. A similar statement holds for rectangles in ℝ^d, d>1. These result follow from similar results for finite-dimensional distributions of Brownian motion and subordination. Mathematics Subject Classification (2000) 30C45. Rodrigo Ba?uelos: R. Ba?uelos was supported in part by NSF grant # 9700585-DMS. Tadeusz Kulczycki: T. Kulczycki was supported by KBN grant 2 P03A 041 22 and RTN Harmonic Analysis and Related Problems, contract HPRN-CT-2001-00273-HARP.     

Two classes of finite semigroups and monoids involving Lucas numbers

The class of finitely presented groups is an extension of the class of triangle groups studied recently. These groups are finite and their orders depend on the Lucas numbers. In this paper, by considering the three presentations

On 3-dimensional asymptotically harmonic manifolds

Let (M,g) be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that M is a hyperbolic manifold of constant sectional curvature , provided M is asymptotically harmonic of constant h > 0. Received: 4 October 2007     

Volume preserving curvature flows in Lorentzian manifolds

Let N be a (n + 1)-dimensional globally hyperbolic Lorentzian manifold with a compact Cauchy hypersurface ${\mathcal{S}_{0}}$ and F a curvature function, either the mean curvature H, the root of the second symmetric polynomial ${{\sigma}_{2}=\sqrt{H_{2}}}$ or a curvature function of class (K*), a class of curvature functions which includes the nth root of the Gaussian curvature ${{\sigma}_{n}= K^{\frac{1}{n}}}$ . We consider curvature flows with curvature function F and a volume preserving term and prove long time existence of the flow and exponential convergence of the corresponding graphs in the C ^∞-topology to a hypersurface of constant F-curvature, provided there are barriers. Furthermore we examine stability properties and foliations of constant F-curvature hypersurfaces.     

A Markoff-type chain for the view-obstruction problem for spheres in ℝ4

In the view-obstruction problem, congruent, closed, convex bodies centred at the points in ⁿ are expanded uniformly until they block all rays from the origin into the open positive cone. The central problem is to determine the minimal blocking size. In the case of spheres of diameter 1, this value is denoted byv(n) and is known for dimensionsn=2,3. Here we show that and obtain a Markoff type chain of isolated extreme values.     

Complete Hypersurfaces with Constant Mean Curvature and Finite Total Curvature

The main result of this paper states that the traceless second fundamental tensor A⁰ of an n-dimensional complete hypersurface M, with constant mean curvature H and finite total curvature, _M |A⁰|ⁿ dv_M < , in a simply-connected space form (c), with non-positive curvature c, goes to zero uniformly at infinity. Several corollaries of this result are considered: any such hypersurface has finite index and, in dimension 2, if H ² + c > 0, any such surface must be compact.     

On the classification of 4-dimensional $$(m,\rho )$$-quasi-Einstein manifolds with harmonic Weyl curvature

In this paper we study four-dimensional \((m,\rho )\)-quasi-Einstein manifolds with harmonic Weyl curvature when \(m\notin \{0,\pm 1,-2,\pm \infty \}\) and \(\rho \notin \{\frac{1}{4},\frac{1}{6}\}\). We prove that a non-trivial \((m,\rho )\)-quasi-Einstein metric g (not necessarily complete) is locally isometric to one of the following: (i) \({\mathcal {B}}^2_\frac{R}{2(m+2)}\times {\mathbb {N}}^2_\frac{R(m+1)}{2(m+2)}\), where \({\mathcal {B}}^2_\frac{R}{2(m+2)}\) is the northern hemisphere in the two-dimensional (2D) sphere \({\mathbb {S}}^2_\frac{R}{2(m+2)}\), \({\mathbb {N}}_\delta \) is a 2D Riemannian manifold with constant curvature \(\delta \), and R is the constant scalar curvature of g. (ii) \({\mathcal {D}}^2_\frac{R}{2(m+2)}\times {\mathbb {N}}^2_\frac{R(m+1)}{2(m+2)}\), where \({\mathcal {D}}^2_\frac{R}{2(m+2)}\) is half (cut by a hyperbolic line) of hyperbolic plane \({\mathbb {H}}^2_\frac{R}{2(m+2)}\). (iii) \({\mathbb {H}}^2_\frac{R}{2(m+2)}\times {\mathbb {N}}^2_\frac{R(m+1)}{2(m+2)}\). (iv) A certain singular metric with \(\rho =0\). (v) A locally conformal flat metric. By applying this local classification, we obtain a classification of the complete \((m,\rho )\)-quasi-Einstein manifolds given the condition of a harmonic Weyl curvature. Our result can be viewed as a local classification of gradient Einstein-type manifolds. A corollary of our result is the classification of \((\lambda ,4+m)\)-Einstein manifolds, which can be viewed as (m, 0)-quasi-Einstein manifolds.     

Uniformly Accurate Quantile Bounds for Sums of Arbitrary Independent Random Variables

Fix any n≥1. Let X ₁,…,X _n be independent random variables such that S _n =X ₁+⋅⋅⋅+X _n , and let S^*_n=sup_{1 ￡ k ￡ n}S_kS^{*}_{n}=\sup_{1\le k\le n}S_{k} . We construct upper and lower bounds for s _y and s_y^*s_{y}^{*} , the upper \frac1y\frac{1}{y} th quantiles of S _n and S^*_nS^{*}_{n} , respectively. Our approximations rely on a computable quantity Q _y and an explicit universal constant γ _y , the latter depending only on y, for which we prove that

A speciality theorem for curves in P<Superscript>5</Superscript>

Let be an integral projective curve. One defines the speciality index e(C) of C as the maximal integer t such that , where ω _C denotes the dualizing sheaf of . Extending a classical result of Halphen concerning the speciality of a space curve, in the present paper we prove that if is an integral degree d curve not contained in any surface of degree  < s, in any threefold of degree  < t, and in any fourfold of degree  < u, and if , then Moreover equality holds if and only if C is a complete intersection of hypersurfaces of degrees u, , and . We give also some partial results in the general case , .