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Register a new userInjectivity in higher order complex domains
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Authors
Naveen Gupta
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This is an expository survey of the Jacobian problem for the class of Pluriharmonic functions.
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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 $mathbb C^n$.
Define a subset of the complex plane to be a Rolles domain if it contains (at least) one critical point of every complex polynomial P such that P(-1)=P(1). Define a Rolles domain to be minimal if no proper subset is a Rolles domain. In this paper, we investigate minimal Rolles domains.
We show that the efficiency of a natural pairing between certain projectively invariant Hardy spaces on dual strongly C-linearly convex real hypersurfaces in complex projective space is measured by the norm of the corresponding Leray transform.
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.
In 2009, the first author introduced a class of zeta functions, called `distance zeta functions, which has enabled us to extend the existing theory of zeta functions of fractal strings and sprays (initiated by the first author and his collaborators in the early 1990s) to arbitrary bounded (fractal) sets in Euclidean spaces of any dimensions. A closely related tool is the class of `tube zeta functions, defined using the tube function of a fractal set. These zeta functions exhibit deep connections with Minkowski contents and upper box (or Minkowski) dimensions, as well as, more generally, with the complex dimensions of fractal sets. In particular, the abscissa of (Lebesgue, i.e., absolute) convergence of the distance zeta function coincides with the upper box dimension of a set. We also introduce a class of transcendentally quasiperiodic sets, and describe their construction based on a sequence of carefully chosen generalized Cantor sets with two auxilliary parameters. As a result, we obtain a family of maximally hyperfractal compact sets and relative fractal drums (i.e., such that the associated fractal zeta functions have a singularity at every point of the critical line of convergence). Finally, we discuss the general fractal tube formulas and the Minkowski measurability criterion obtained by the authors in the context of relative fractal drums (and, in particular, of bounded subsets of the N-dimensional Euclidean space).
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