22 research outputs found
Polya's inequalities, global uniform integrability and the size of plurisubharmonic lemniscates
First we prove a new inequality comparing uniformly the relative volume of a
Borel subset with respect to any given complex euclidean ball \B \sub \C^n
with its relative logarithmic capacity in \C^n with respect to the same ball
\B.
An analoguous comparison inequality for Borel subsets of euclidean balls of
any generic real subspace of \C^n is also proved.
Then we give several interesting applications of these inequalities.
First we obtain sharp uniform estimates on the relative size of \psh
lemniscates associated to the Lelong class of \psh functions of logarithmic
singularities at infinity on \C^n as well as the Cegrell class of
\psh functions of bounded Monge-Amp\`ere mass on a hyperconvex domain \W
\Sub \C^n.
Then we also deduce new results on the global behaviour of both the Lelong
class and the Cegrell class of \psh functions.Comment: 25 page
A minimum principle for plurisubharmonic functions
The main goal of this work is to give new and precise generalizations to
various classes of plurisubharmonic functions of the classical minimum modulus
principle for holomorphic functions of one complex variable, in the spirit of
the famous lemma of Cartan-Boutroux. As an application we obtain precise
estimates on the size of "plurisubharmonic lemniscates" in terms of appropriate
Hausdorff contents
On the average number of zeros of random harmonic polynomials with i.i.d. coefficients: precise asymptotics
Addressing a problem posed by W. Li and A. Wei (2009), we investigate the
average number of (complex) zeros of a random harmonic polynomial sampled from the Kac ensemble, i.e., where the coefficients
are independent identically distributed centered complex Gaussian random
variables. We establish a precise asymptotic, showing that when tends to infinity the average number of zeros is asymptotic
to . We further consider the average number of zeros
restricted to various regions in the complex plane leading to interesting
comparisons with the classically studied case of analytic Kac polynomials. We
also consider deterministic extremal problems for harmonic polynomials with
coefficient constraints; using an indirect probabilistic method we show the
existence of harmonic polynomials with unimodular coefficients having at least
zeros. We conclude with a list of open
problems.Comment: 27 pages, 1 figur
Constructive Function Theory on Sets of the Complex Plane through Potential Theory and Geometric Function Theory
This is a survey of some recent results concerning polynomial inequalities
and polynomial approximation of functions in the complex plane. The results are
achieved by the application of methods and techniques of modern geometric
function theory and potential theory
A 1-parameter family of spherical CR uniformizations of the figure eight knot complement
We describe a simple fundamental domain for the holonomy group of the
boundary unipotent spherical CR uniformization of the figure eight knot
complement, and deduce that small deformations of that holonomy group (such
that the boundary holonomy remains parabolic) also give a uniformization of the
figure eight knot complement. Finally, we construct an explicit 1-parameter
family of deformations of the boundary unipotent holonomy group such that the
boundary holonomy is twist-parabolic. For small values of the twist of these
parabolic elements, this produces a 1-parameter family of pairwise
non-conjugate spherical CR uniformizations of the figure eight knot complement