Showing papers in "Tohoku Mathematical Journal in 1983"

Generic properties of the eigenvalue of the laplacian for compact riemannian manifolds

Abstract: Introduction. In this paper, we discuss generic properties of the eigenvalues of the Laplacian for compact Riemannian manifolds without boundary. Throughout this paper, let M be an arbitrary fixed connected compact C°° manifold of dimension n without boundary, and ^t the set of all C°° Riemannian metrics on M. For g e^t, let Δg be the Laplacian (cf. (2.1)) of (M, g) acting on the space C°°(M) of all C°° real valued functions on M and 0 = λo(flf) < λΛflO ^ x2(g) ^ T°° the eigenvalues of the Laplacian Δg counted with their multiplicities. We regard each eigenvalue Xk(g), k = 0,1, 2, , as a function of g in ^. Let us consider the following problem: "Does each eigenvalue Xk(g) depend continuously on g in ^t with respect to the C°° topology*!" The continuous dependence of the eigenvalues of the Dirichlet problem upon variations of domains is well known (cf. [CH, p. 290]). Variations of coefficients of elliptic differential operators were dealt with by KodairaSpencer [KS] who gave a proof of the continuity of eigenvalues. In this paper, we give a simple proof of the above problem. To answer the above problem, in § 1, we introduce a complete distance p on ^t which gives the C°° topology. Then, in § 2, we assert that each χk(g)f k = 1, 2, , depends continuously on g e ^t with respect to the topology on Λ? induced by the distance p. More precisely, we have

Higher-dimensional analogues of periodic continued fractions and cusp singularities

Abstract: Introduction. There is a well-known relationship between periodic continued fractions and 2-dimensional cusp singularities. (See, for instance, [10], [11].) Let π: U→V be the minimal resolution of a 2dimensional cusp singularity (V,p). Then the exceptional set X=π-1(p) is either a cycle of s rational curves with self-intersection numbers a1, a2, ..., as≦-2 at least one of which is strictly smaller than -2(s≧2), or a rational curve with a node and with a self-intersection number a<0. Then we can associate to it the periodic continued fraction

Convergence of positive linear approximation processes

Abstract: Soit X un espace de Hausdorff compact et soit B(X) le treillis de Banach de toutes les fonctions bornees a valeur reelle sur X avec la norme sup. II•II C(X) est le sous treillis de B(X). On etablit un theoreme de type Korovkin

Further remarks on the exponent of convergence of Poincaré series

Abstract: group Con(n) of (n+1)-dimensional hyperbolic space Hn+1 We shall associate in • ̃3 a certain number ƒÂ(G), 0•...ƒÂ(G)•...n, to G called the exponent of convergence of the Poincare series attached to G. It is related to several of the geometric properties of G; these properties have been the subject of many investigations but in [5] Sullivan has discussed these exhaustively and completed them in several important points. The question with which this paper is concerned is that of estimating

Stability properties for functional differential equations with infinite delay

Abstract: On considere des equations differentielles fonctionnelles a retard infini pour lesquelles on sait que l'operateur solution n'est pas necessairement completement continu mais est une α-contraction

Totally geodesic foliations and Killing fields

Abstract: Soit (M,g) une variete de Riemann connexe fermee et #7B-F un feuilletage totalement geodesique de codimension un de (M,g). Alors tout champ de Killing Z sur (M,g) preserve #7B-F: le flot Z applique chaque feuillet de #7B-F sur un feuillet de #7B-F

The local homology of cut loci in Riemannian manifolds

Abstract: point q-and the position of q in the simplicial decomposition. Cut points of order one are extreme points of the simplicial complex; those of order two are interior to an edge; and those of order k•†3 are vertices where k edges meet. In short, the topology of a neighborhood of a point in the cut locus is determined by the order of the cut point. Recently, Ozols [12] and Buchner [3] have shown that the cut locus of a point p in a real analytic Riemannian manifold admits a simplicial decomposition. Moreover, Ozols [12] describes the structure of the cut locus near a non-conjugate cut point q as a finite (depending on the order of q) intersection of hyperspaces and half-planes, while Buchner [4] completely classifies the local structure of generic cut loci in low dimensional manifolds. In general, however, the relation between the set, called the link, of minimal geodesics connecting p to a cut point q and the structure of the cut locus near q remains obscure. This paper establishes, as a consequence of Poincare duality, a duality between the Cech cohomology of the link and the local homology groups of cut loci in smooth Riemannian manifolds, thereby weakly generalizing the result of Myers on the order and local topology of cut loci in real analytic surfaces. Using standard arguments from algebraic topology, we show that certain local homology groups of cut loci are torsion free. Finally, we prove interconnections between the dimension of the cut locus and the vanishing of high dimensional local homology which lead up to a generalization of a theorem of Bishop [2] on the decomposition of cut loci.

Comparison method and stability problem in functional differential equations

Abstract: In this paper, using the comparison method and borrowing the ideas and terminologies from Kato [1], [2], [3], we discuss the stability in functional differential equations with infinite delay. We also give some extensions of the ideas in [5], [6], [7]. As a corollary to our results, the corresponding stability theorem of Kato [1] is included. Let X be a linear space of ^\"-valued functions on ( — °° , 0] with a semi-norm \\\\ \\\\x, and denote by Xτ the space of functions φ(s) on (— °°, 0] which are continuous on [ — τ, 0] and satisfying φ_τeX for τ ^ 0, where and henceforth φt denotes the function on (— ̂ ,0] defined by φt(s) = Φ(t + 8). The space X is said to be admissible, if the following are satisfied: For any τ ^ 0 and any φ e Xτ (a) φteX for all te[—τ, 0], especially, φQ = φ e X; (b) φt is continuous in te[ — τ, 0]; (c) μ || 0(0) | |^| |0 |L ^ K(τ) sup_r,8,0 1| φ(s) \\\\ + M(τ) \\\\ φ_τ \\\\Z9 where μ > 0 is a constant and K(τ)9 M(τ) are continuous. Consider the functional differential equation (E) * = /(ί, xt) and assume that f(t, 0) = 0 and that f(t, φ) is completely continuous on I x X where X is an admissible space and /= [0, oo). For the fundamental properties of the solutions of (E), we refer to [4], Let Y be an admissible space satisfying XdY and where N > 0 is a constant. Let x(t) be an arbitrary solution of (E). The definitions of stability in (X, Y) will be given as follows: The zero solution of (E) is said to be ( i ) stable in (X, Y), if for any ε > 0 and any τ ^ 0, there is a δ = δ(τ, ε) > 0 such that ||a?r||z.< d implies \\\\xt\\\\γ

Products of distributions in $H^{p}$ spaces

Abstract: Si K 1 , ..., Kn sont des operateurs integraux singuliers de type Calderon-Zygmund sur R n et Si et si p, q, r>0 satisfont 1/p=1/q+1/r<1+N/n, alors il y a une constante C dependant seulement de p,q,r,K 1 ,...,Kn et n, telle que pour tout h∈φ∩ q (Rn) et tout g∈φH2(Rn), ∥SJ(−1) 151 (Π I∈J K J )h•(Π j∈jc K j )g∥ HP ≤C∥h∥ Hq ∥g∥H r , ou la sommation est sur tous les sous-ensembles J de {1,...,N}

Approximation by partial sums and Cesàro means of multiple orthogonal series

Abstract: Soit (X,F,μ) un espace de mesure positive arbitraire et {φ ik (x):i,k=1,2,...} un systeme orthonormal sur cet espace. On considere la serie orthogonale double Σ i=1 ∞ Σ k=1 ∞ a ik φ ik (x), ou {a ik :i,k=1,2,...} est une suite double de nombres reels