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Register a new userSmooth equivalence of deformations of domains in complex euclidean spaces
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We prove that two smooth families of 2-connected domains in $cc$ are smoothly equivalent if they are equivalent under a possibly discontinuous family of biholomorphisms. We construct, for $m geq 3$, two smooth families of smoothly bounded $m$-connected domains in $cc$, and for $ngeq2$, two families of strictly pseudoconvex domains in $cc^n$, that are equivalent under discontinuous families of biholomorphisms but not under any continuous family of biholomorphisms. Finally, we give sufficient conditions for the smooth equivalence of two smooth families of domains.
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In this paper we find big Euclidean domains in complex manifolds. We consider open neighbourhoods of sets of the form $Kcup M$ in a complex manifold $X$, where $K$ is a compact $mathscr O(U)$-convex set in an open Stein neighbourhood $U$ of $K$, $M$ is an embedded Stein submanifold of $X$, and $Kcap M$ is compact and $mathscr O(M)$-convex. We prove a Docquier-Grauert type theorem concerning biholomorphic equivalence of neighbourhoods of such sets, and we give sufficient conditions for the existence of Stein neighbourhoods of $Kcup M$, biholomorphic to domains in $mathbb C^n$ with $n=dim X$, such that $M$ is mapped onto a closed complex submanifold of $mathbb C^n$.
The second named author and David Kalaj introduced a pseudometric on any domain in the real Euclidean space $mathbb R^n$, $nge 3$, defined in terms of conformal harmonic discs, by analogy with Kobayashis pseudometric on complex manifolds, which is defined in terms of holomorphic discs. They showed that on the unit ball of $mathbb R^n$, this minimal metric coincides with the classical Beltrami-Cayley-Klein metric. In the present paper we investigate properties of the minimal pseudometric and give sufficient conditions for a domain to be (complete) hyperbolic, meaning that the minimal pseudometric is a (complete) metric. We show in particular that a domain having a negative minimal plurisubharmonic exhaustion function is hyperbolic, and a bounded strongly minimally convex domain is complete hyperbolic. We also prove a localization theorem for the minimal pseudometric. Finally, we show that a convex domain is complete hyperbolic if and only if it does not contain any affine 2-plane.
We introduce a class of normal complex spaces having only mild sin-gularities (close to quotient singularities) for which we generalize the notion of a (analytic) fundamental class for an analytic cycle and also the notion of a relative fundamental class for an analytic family of cycles. We also generalize to these spaces the geometric intersection theory for analytic cycles with rational positive coefficients and show that it behaves well with respect to analytic families of cycles. We prove that this intersection theory has most of the usual properties of the standard geometric intersection theory on complex manifolds, but with the exception that the intersection cycle of two cycles with positive integral coefficients that intersect properly may have rational coefficients. AMS classification. 32 C 20-32 C 25-32 C 36.
We study the stability of John domains in Banach spaces under removal of a countable set of points. In particular, we prove that the class of John domains is stable in the sense that removing a certain type of closed countable set from the domain yields a new domain which also is a John domain. We apply this result to prove the stability of the inner uniform domains. Finally, we consider a wider class of domains, so called $psi$-John domains and prove a similar result for this class.
This is an expository survey of the Jacobian problem for the class of Pluriharmonic functions.
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