50 research outputs found
Semigroups of operators on Hardy spaces and cocycles of flows
No full text1 online resource (PDF, 10 pages)Jafari, Farhad; Slodkowski, Zbigniew; Tonev, Thomas. (2008). Semigroups of operators on Hardy spaces and cocycles of flows. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/179961
Some properties of Grauert type surfaces
No full textIn a previous work, we classified weakly complete surfaces which admit a real analytic plurisubharmonic exhaustion function; we showed that, if they are not proper over a Stein space, then they admit a pluriharmonic function, with compact Levi-flat level sets foliated with dense complex leaves. We called these Grauert type surfaces. In this note, we investigate some properties of these surfaces. Namely, we prove that the only compact curves that can be contained in them are negative in the sense of Grauert and that the level sets of the pluriharmonic function are connected, thus completing the analogy with the CartanâRemmert reduction of a holomorphically convex space. Moreover, in our classification theorem, we had to pass to a double cover to produce the pluriharmonic function; the last part of the present paper is devoted to the construction of an example where passing to a double cover cannot be avoided
On weakly complete surfaces [Sur les surfaces faiblement complètes]
No full textA weakly complete space is a (connected) complex space endowed with a (smooth) plurisubharmonic exhaustion function. In this paper, we classify the weakly complete surfaces (i.e. weakly complete manifolds of dimension 2) for which such exhaustion function can be chosen to be real analytic: they can be modifications of Stein spaces or proper (i.e. endowed with a proper surjective holomorphic map onto) a non-compact (possibly singular) complex curve or surfaces of Grauert type i.e. foliated with real analytic Levi flat hypersurfaces whose Levi foliation has dense complex leaves. In the last case, we also show that such Levi flat hypersurfaces are in fact level sets of a global proper pluriharmonic function, up to passing to a holomorphic double covering.Un espace complexe est dit faiblement
complet s'il est muni d'une fonction d'exhaustion plurisousharmonique. Dans ce
papier on classie les surfaces complexes faiblement complètes qui admettent une
fonction d'exhaustion plurisousharmonique et analytique réelle. Elles sont des types
suivants : modications des espaces de Stein, surfaces complexes propres sur des
courbes complexes non compactes, ou bien surfaces complexes de type Grauert
i.e. feuilletées par des hypersurfaces Levi plates dont les feuilles du feuilletage de
Levi sont partout denses. Dans ce dernier cas on montre aussi que, sauf à passer à
un double revêtement, les hypersurfaces Levi plates sont en fait les niveaux d'une
fonction pluriharmonique globale
Weakly complete complex surfaces
No full textA weakly complete space is a complex space that admits a (smooth) plurisubharmonic exhaustion function. In this paper, we classify those weakly complete complex surfaces for which such an exhaustion function can be chosen to be real analytic: they can be modifications of Stein spaces or proper over a non-compact (possibly singular) complex curve, or foliated with real-analytic Levi flat hypersurfaces which in turn are foliated by dense complex leaves (these we call "surfaces of Grauert type"). In the last case, we also show that such Levi flat hypersurfaces are in fact level sets of a global proper pluriharmonic function, up to passing to a holomorphic double cover of the space. Our method of proof is based on the careful analysis of the level sets of the given exhaustion function and their intersections with the \emph{minimal singular set}, that is, the set where every plurisubharmonic exhaustion function has a degenerate Levi form
Domains with a continuous exhaustion in weakly complete surfaces
No full textIn previous works, Tomassini and the authors studied and classified complex surfaces admitting a real-analytic plurisubharmonic exhaustion function; let X be such a surface and D⊆ X a domain admitting a continuous plurisubharmonic exhaustion function: what can be said about the geometry of D? If the exhaustion of D is assumed to be smooth, the second author already answered this question; however, the continuous case is more difficult and requires different methods. In the present paper, we address such question by studying the local maximum sets contained in D and their interplay with the complex geometric structure of X; we conclude that, if D is not a modification of a Stein space, then it shares the same geometric features of X
Local maximum property and q-plurisubharmonic functions in uniform algebras
No full textAbstractThere is proven a formula expressing higher order Shilov boundaries of the tensor products of uniform algebras in terms of boundaries of the factor algebras. Main step: Cartesian product of k- and l-maximum sets is a (k + l + 1)-maximum set. (Definition: locally closed X⊂Cn is a k-maximum set if polynomials have local maximum property on intersections of X with (n-k)-dimensional complex planes.) Other properties and characterizations of k-maximum sets are given, e.g., X⊂Cn is a k-maximum set iff each k-plurisubharmonic function has local maximum property on X
Polynomial hulls with convex fibers and complex geodesics
No full textAbstractLet X be a compact subset of {|z| = 1} × Cn with convex fibers. Several equivalent conditions are obtained that characterize polynomially convex hulls Y = X̂ with nonempty interior. Under the latter assumption, it is shown that every (z0, w0) ϵ ∂ Y and such that ¦z0¦ < 1 is contained in a closed n-dimensional complex submanifold of d × Cn which is tangent to Y along an analytic disc.We show, as an application, that some results of Lempert on complex geodesies in convex domains are direct consequences of properties of polynomial hulls
Pseudoconvex classes of functions. III. Characterization of dual pseudoconvex classes on complex homogeneous spaces
No full textInvariant classes of functions on complex homogeneous spaces, with properties similar to those of the class of plurisubharmonic functions, are studied. The main tool is a regularization method for these classes, and the main theorem characterizes dual classes of functions (where duality is defined in terms of the local maximum property). These results are crucial in proving a duality theorem for complex interpolation of normed spaces, which is given elsewhere.</p
