We construct examples of non-locally embeddable structures. These examples may show some improvement on previous examples by Nirenberg, and Jacobowitz and Trèves. They are based on a simple construction which consists in gluing two embedded structures. And (this is our main point) we believe that these examples are very transparent, therefore easy to work with.
The study of -holomorphic maps leads to the consideration of the inequations , and . The first inequation is fairly easy to use. The second one, that is relevant to the case of rough structures, is more delicate. The case of vector valued is strikingly different from the scalar valued case. Unique continuation and isolated zeroes are the main topics under study. One of the results is that, in almost complex structures of Hölder class , any -holomorphic curve that is constant on a non-empty...
Holomorphic bundles, with fiber , defined on open sets in by locally constant transition automorphisms, are shown to extend to holomorphic bundles on the Riemann sphere. In particular, it allows us to give an example of a non-Stein holomorphic bundle on the unit disc, with polynomial transition automorphisms.
Dans quelle mesure une algèbre uniforme est-elle déterminée par l’espace vectoriel des parties réelles de ses éléments ? On s’intéresse à ce problème pour des algèbres définies sur des sous-ensembles compacts du plan complexe de connectivité finie.
Nous prouvons en particulier que tout domaine homogène borné de , à frontière deux fois continûment différentiable est bi-holomorphiquement équivalent à la boule unité de . Les démonstrations sont entièrement élémentaires.
This paper is an outgrowth of a paper by the first author on a generalized Hartogs Lemma. We complete the discussion of the nonlinear ∂̅ problem ∂f/∂z̅ = ψ(z,f(z)). We also simplify the proofs by a different choice of Banach spaces of functions.
We study sets in the boundary of a domain in , on which a holomorphic function has maximum modulus. In particular we show that in a real analytic strictly pseudoconvex boundary, maximum modulus sets of maximum dimension are real analytic. Maximum modulus sets are related to , which are sets along which appropriate collections of holomorphic and antiholomorphic functions agree.
This note contains an approximation theorem that implies that every compact subset of is a good compact set in the sense of Martineau. The property in question is fundamental for the extension of analytic functionals. The approximation theorem depends on a finiteness result about certain polynomially convex hulls.
The notion of “strong boundary values” was introduced by the authors in the local theory of hyperfunction boundary values (boundary values of functions with unrestricted growth, not necessarily solutions of a PDE). In this paper two points are clarified, at least in the global setting (compact boundaries): independence with respect to the defining function that defines the boundary, and the spaces of test functions to be used. The proofs rely crucially on simple results in spectral asymptotics.
The definition of the Kobayashi-Royden pseudo-metric for almost complex manifolds is similar to its definition for complex manifolds. We study the question of completeness of some domains for this metric. In particular, we study the completeness of the complement of submanifolds of co-dimension 1 or 2. The paper includes a discussion, with proofs, of basic facts in the theory of pseudo-holomorphic discs.
We give a sufficient condition for a hermitian holomorphic vector bundle over the disk to be quasi-isometric to the trivial bundle. One consequence is a version of Cartan's lemma on the factorization of matrices with uniform bounds.
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