Showing papers in "Mathematische Annalen in 1987"

Fibered Algebraic Surfaces with Low Slope

Abstract: The background of this paper is the following conjecture on complex surfaces of general type: Let S be a (smooth, projective) minimal surface of general type over ti? with c~ 2 throughout the paper.

Scalar Curvature of a Metric with Unit Volume

Abstract: The problem of finding Riemannian metrics on a closed manifold with prescribed scalar curvature function is now fairly well understood from the works of Kazdan and Warner in 1970's ([10] and references cited in it). In this paper we shall consider the same problem under a constraint on the volume. For this purpose it is useful to introduce an invariant p(M) of a smooth closed manifold M, which will be called the Yamabe number of M, defined as the supremum of #(M, C) of all conformal classes C of Riemannian metrics on M,

Isoparametric functions on Riemannian manifolds. I

Abstract: On etablit les proprietes de base des fonctions transnormales et on demontre le lemme du tube. On demontre que: soit M une variete de Riemann lisse, complete, connexe et f une fonction transnormale sur M. Alors: a) les varietes focales de f sont des sous-varietes lisses de M; b) chaque ensemble de niveau regulier de f est un tube sur d'autres des varietes focales

The gromov norm of the Kaehler class of symmetric domains

Abstract: Let D be bounded symmetric domain of rank p, and let o9 be the KRhler form of its Bergmann metric, where the metric is normalized so that the minimum holomorphic sectional curvature is 1. I fX is a compact manifold whose universal cover is D, to defines a cohomology class [to] e H 2 (X, R). The purpose of this paper is to compute its sup norm in the sense of Gromov [5]. We show II [ to] II co = pn. Strictly speaking, we only prove this for three of the four classes of classical domains (cf. Sect. 2). This theorem has the following topological corollary. Let S be a Riemann surface of genus 0 > 1 and f : S ~ X a continuous map. Then ! f ' t o < 4 p ( # 1)n.

On Some Operator Inequalities.

Abstract: A real-valued continuous function f(2) defined on a (finite or infinite) interval of the real line is said to be operator monotone if, for any pair of bounded selfadjoint operators A, B on an infinite dimensional Hilbert space, A > B implies f (A) > f(B). Here f(A) and f(B) are defined according to the usual functional calculus, and > refers to the order relation induced by the cone of positive (semi-definite) operators. Operator monotony is quite a restrictive condition. In fact, for p > 0 the function f(2): = 2 p is operator monotone on the half-line [0, c~) if and only if 0 1 the function f(2) = 2 p possesses some property akin to operator monotony. In this direction Furuta I-3] recently observed that if r > 0, p > 0, q > 1 satisfies (1 + 2r)q > p + 2r then A > B > 0 implies A tp § 2,)/q > (ArBPA r) l/~. If p = 2r and q = 2, the requirement is automatically satisfied, hence A p > (Ap/2BPAP/2)I/2 whenever A > B > 0. When A is invertible, this means that

Localization of the Kobayashi Metric and the Boundary Continuity of Proper Holomorphic Mappings

Abstract: A classical theorem of Carath6odory [8] states that every biholomorphic map f: D~ ~D2 between domains in the complex plane C bounded by simple closed Jordan curves extends to a homeomorphism of/31 onto/ )2 . There are some well-known generalizations of this result to domains in IE a. If D t and D2 are bounded pseudoconvex domains in 112" with if2 boundary and if the infinitezimal Kobayashi metric on D2 grows sufficiently fast near the boundary of D2(Ko2(g; X) >= [Xl/d(z, bDz) ~ for some e e (0, 1)), then every proper holomorphic map f : Dt--*D2 extends to a H61der-continuous map of/51 onto/32 [2, 3, 10, 25, 29-32]. This holds in particular if/92 is strictly pseudoconvex or if it is pseudoconvex with real-analytic boundary. The same result holds if D2 is only piecewise smooth strongly pseudoconvex [31]. Further results treat exceptional cases such as the balls [1 ], Reinhardt domains [5, 24], domains with many symmetries [4], and analytically bounded Hartogs domains in C 2 [I1]. Biholomorphic maps between certain types of nonpseudoconvex domains with real-analytic boundaries were treated in [13] and [27]. Besides the papers mentioned above there is a vast literature concerning smooth extension of proper holomorphic maps of smoothly bounded pseudoconvex domains ([6, 7, 12, 14], to mention just a few). In the present paper we obtain some results on the continuous extension of proper holomorphic maps f : D1--*/)2 under local assumptions on the boundaries bD1 and bD2. One of the main results is

Arithmetic distance functions and height functions in diophantine geometry

Abstract: In virtually all areas of Diophantine Geometry, the theory of height functions plays a crucial role. This theory associates to each (Cartier) divisor D on a projective variety V a function ho mapping the group of rational points of V to the real numbers. This function, which is defined up to a bounded function on V, has very nice functorial properties. For example, it is linear in D and depends only on the linear equivalence class of D. As a consequence, any relation between divisor classes, such as the theorem of the square for abelian varieties, will yield a corresponding relation for height functions. Further, it is possible to write the height function ho as a sum of local height functions 20(" ; v), where v ranges over the distinct absolute values of the given field. (These local heights are also known as logarithmic distance functions or Weil functions.) Each 2 o is defined away from the support of D, and gives a measure of the v-adic distance from the given point of V to the divisor D. In this paper we propose to define local height functions 2 x for every closed subscheme X of a projective variety V. As above, 2x(; v) will give a measure of the v-adic distance from a point of V to X. These functions will have many nice functorial properties; for example, 2xny will equal the minimum of 2x and hr. In this way, relations between closed subschemes (or equivalently, between ideal sheaves) will yield relations between local height functions. We will thus be able to convert geometric statements into arithmetic statements relatively painlessly. More generally, we will deal with the case that V is merely quasi-projective by defining a function 20v('; v) which will measure the v-adic distance to the "boundary" of V. We will then be able to assign to each closed subscheme X of V a local height function 2x which will be well defined up to addition of a multiple of 2~v. This extra generality is useful, for example, when one has a complete family of varieties and wishes to discard the "bad" fibers. As one application of the machinery that we have developed, we prove a quantitative version of the inverse function theorem. Thus given a finite map, we use local height functions to describe how far away from the ramification locus

Systèmes dynamiques non commutatifs et moyennabilité.

Abstract: On introduit la notion de fonction de type positif relativement a un systeme dynamique C*, dont les proprietes sont analogues a celles des fonctions de type positif classiques. On montre que dans le cas d'un groupe discret G agissant sur une algebre de Von Neumann M, la moyennabilite de l'action est equivalente a l'existence d'une suite bornee de fonctions de type positif sur G a support fini, a valeurs dans le centre Z(M) de M qui converge vers 1 pour la topologie de la convergence simple ultrafaible

Perturbation theory for dual semigroups I. The sun-reflexive case

Abstract: Soit A 0 le generateur infinitesimal d'un Co-semigroupe d'operateurs lineaires bornes To(t) sur un espace de Banach X, soit B un operateur lineaire borne sur X, A 0 +b engendre un Co-semigroupe T(t) sur X. B agit hors de l'espace X dans un espace plus grand Y. On donne une methode systematique pour construire un tel espace Y

Boundaries of a convex set and interpolation sets

Abstract: The structure of the "boundary" of a convex set C is a field of intensive research in functional analysis. The classical "boundary" is the set Ext(C) of the extreme points of C at least when some compactness is assumed and C is recovered from Ext(C) by means of the integral representation theory. In this paper, a more general notion of boundary (Definition 1.I) is considered. Such a boundary needs not contain, or even meet, the extreme points, and thus the classical tools are not available. However, through R. C. James's technique and a remarquable result of S. Simons, a convex set can often be "recovered", in a strong sense, from its boundary (Sect. 1). Tight connections are established between this notion, lacunarity in harmonic analysis (Sect. 2), Banach spaces containing or not ?(N) (Sect. 3), compactness and duality theory (Sect. 4). Finally, Sect. 5 is devoted to miscellaneous results and questions in the spirit of the work, and to examples showing the necessity of the assumptions we made. As a general remark, we did not systematically try to state the results in their most general form, as far as it was not necessary to introduce new ideas for doing so. This is specially true for the results of Sect. 2, where we worked in the duality (2~,~?) of the "little" Fourier analysis although the techniques we used remain valid within the general frame of locally compact abelian groups.

Unfoldings in knot theory

Abstract: In IN-R] the link at infinity of a hypersurface V CIE r~ defined by a "good" polynomial map f : tY~--.II2 was used as an extended example, f was called "good" if it had only isolated singularities and it was claimed that the link at infinity then always has a "Milnor fibration." This is incorrect (although we have found it to be a common misconception). To correct it, the definition of"good" must be modified as follows.

Scalar Curvatures on Sn.

Abstract: Soit (M n ,g) une variete de Riemann compacte de dimension n≥3, et R(x) une fonction lisse sur M n . On etudie si R(x) peut etre la courbure scalaire d'une metrique g~ qui est conforme point par point a la metrique originale g

Positivity and regularity of hyperbolic Volterra equations in Banach spaces

Abstract: Soit X un espace de Banach, A un operateur lineaire clos non borne dans X a domaine dense D(A), a∈L loc 1 (R + ), f∈C(J, X) avec J=[0, T] et on considere l'equation de Volterra de type scalaire: u(t)=f(t)+∫ 0 t a(t-τ)Au(τ)dτ, t∈J. On etudie l'existence, la positivite, la regularite et la compacite ainsi que l'integrabilite de l'operateur resolvant S(t) defini par: S(t)x=x+∫ 0 t a(t-τ)AS(τ)xdτ, t≥0, x∈D(A) pour deux classes particulieres de noyaux a et d'operateurs A