Ombres Convexité, régularité et sous-harmonicité

Letf:V→R be a function defined on a subsetV ofR ⁿ ×R ^d let?:x→inf{f(x t);t such that(x t)∈V} denote theshadow off and letΦ={(x t)∈V; f(x t)=?(x)} This paper deals with the characterization of some properties of ? in terms of the infinitesimal behavior off near points ζ∈Φ proving in particular a conjecture of J M Trépreau concerning the cased=1 Characterizations of this type are provided for the convexity the subharmonicity or theC ^{1 1} regularity of ? in the interior ofI={x∈ R ⁿ;εR ^d (x t)∈V} and in theC ^{1 1} case an expression forD ²? is given To some extent an answer is given to the following question: which convex function ?:I→R I interval ?R (resp which function √:I→R of classC ^{1 1}) is the shadow of aC ² functionf:I×R→R?     

ber holomorphe Abbildungen komplexer Räume

Ohne ZusammenfassungHerrn Hermann Schmidt zum 60. Geburtstag gewidmetDer Deutschen Forschungsgemeinschaft danke ich für das Stipendium, das es mir ermöglichte, diese Arbeit anzufertigen.     

Sur la convexité holomorphe des revêtements linéaires réductifs d’une variété projective algébrique complexe

Let X be a smooth connected compact projective variety over . We study the Shafarevich conjecture concerning holomorphic convexity along the lines of Kollárs approach, when the fundamental group of X admits large finite dimensionnal representations. We prove that, given n, the topological covering space of a projective algebraic compact complex manifold X corresponding to the intersection of the kernels of all linear reductive representations of ₁(X) to GL_n() is holomorphically convex. In the surface case, this is a corollary of a theorem due to Katzarkov and Ramachandran.      

Einige Ungleichungen für in einem Parallelstreifen holomorphe Funktionen

Ohne Zusammenfassung     

Über holomorphe Abbildungen projektiv-algebraischer Mannigfaltigkeiten auf komplexe Räume

Ohne ZusammenfassungUnterstützt durch Air Force Office of Scientific Research, AF 49 (638)-253, S/A1.     

Existenzsätze für maximale holomorphe und meromorphe Abbildungen

Ohne Zusammenfassung     

Sur la normalisation holomorphe de structures de Poisson à 1-jet nul

We show that a Poisson structure whose linear part vanish can be holomorphically normalized in a neighbourhood of its singular point $0\in {\mathbb{C}}^n$ if on the one hand, a Diophantine condition on a Lie algebra associated to the quadratic part is satisfied, and, on the other hand, the normal form satisfies some formal conditions. To cite this article: P. Lohrmann, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

L'ensemble des points fixes d'une application holomorphe dans un produit fini de boules-unités d'espaces de Hilbert est une sous-variété banachique complexe

Résumé Dans cet article, nous montrons le résultat suivant. Soit B=B₁×...×B_p un produit fini de boulesunités d'espaces de Hilbert H_i, et soit f: BB une application holomorphe. Alors, l'ensemble des points fixes de f est une sousvariété banachique complexe de B, et est rétracte holomorphe de B.

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Summary In this paper, we prove the following result. Let B=B₁×...×B_p be a finite product of unit balls of complex Hilbert spaces H_i, and let f: BB be a holomorphic map, then the set of fixed points of f is a complex Banach submanifold, and is a holomorphic retract of B.

     

Ein Approximationssatz für holomorphe Funktionen auf nicht-kompakten Riemannschen Flächen

Let R be a non-compact Riemann surface andO(R) the algebra of all holomorphic functions on R. A subalgebraA ?O(R) is calledfull (“voll”), if (F1) for every point ??R there is a function f∈A with a simple zero at ? and no other zeros; (F2) if f, g∈A and f/g has no poles, then f/∈A. In 1971 Ian RICHARDS set the problem whether full subalgebras are dense inO(R), with respect to the topology of compact convergence. We answer this question in the positive, using a lemma of I. RICHARDS and theorems of R. ARENS and the author. Does this approximation theorem remain true for Stein manifolds of dimension n>1, if one modifies condition (F1) in a natural way? A counterexample is provided by a domain of holomorphy G??² and a full, but not dense subalgebraA ?O(G).     

Higher order Lagrange-Poincaré and Hamilton-Poincaré reductions

Motivated by the problem of longitudinal data assimilation, e.g., in the registration of a sequence of images, we develop the higher-order framework for Lagrangian and Hamiltonian reduction by symmetry in geometric mechanics. In particular, we obtain the reduced variational principles and the associated Poisson brackets. The special case of higher order Euler-Poincaré and Lie-Poisson reduction is also studied in detail.     

Approximationssätze für holomorphe Funktionen mit Werten in komplexen Räumen

Ohne ZusammenfassungDie Resultate wurden z. T. in einer C. r.-Note [15] angekündigt. Bei der vorliegenden Publikation handelt es sich um den ersten Teil der Habilitationsschrift des Verf., die in 3 Teilen erscheint (vgl. [16], [17]). Einige Beweise sind jedoch nach Vorschlägen vonH. Cartan abgeändert worden. Ich möchte Herrn ProfessorCartan an dieser Stelle für die freundliche Mitteilung seiner Ideen meinen Dank aussprechen. — Die eckigen Klammern beziehen sich auf das Literaturverzeichnis am Ende der Arbeit.     

Poincaré series

Let N_α denote the number of solutions to the congruence F(x_i,..., x_m) ≡ 0 (mod p^α) for a polynomial F(x_i,..., x_m) with integral p-adic coefficients. We examine the series \(\varphi (t) = \sum\nolimits_{\alpha = 0}^\infty {N_{\alpha ^{t^\alpha } } } \) . called the Poincaré series for the polynomial F. In this work we prove the rationality of the series ?(t) for a class of isometrically equivalent polynomials of m variables, m ≥ 2, containing the sum of two forms ?_n(x, y) + ?_n+1(x, y) respectively of degrees n and n+1, n ≥ 2. In particular the Poincaré series for any third degree polynomial F₃(x, y) (over the set of unknowns) with integral p-adic coefficients is a rational function of t.     

A Poincaré Cone Condition in the Poincaré Group

Ben Arous and Gradinaru (Potential Anal 8(3):217–258, 1998) described the singularity of the Green function of a general sub-elliptic diffusion. In this article we first adapt their proof to the more general context of a hypoelliptic diffusion. In a second time, we deduce a Wiener criterion and a Poincaré cone condition for a relativistic diffusion with values in the Poincaré group (i.e the group of affine direct isometries of the Minkowski space-time).     

Compatibilité d'un fibré de Dirac avec une connexion

We extend to an almost Dirac structure and affine connections the notion of compatibility of a bivectors field or a 2-differential form with a pseudo-metric. Compatibility with a symmetric connection implies integrability. We shall be interested especially by such structures on Lie groups or Riemannian homogeneous spaces.