46 research outputs found
Subextension of plurisubharmonic functions with weak singularities
We prove several results showing that plurisubharmonic functions with various
bounds on their Monge-Ampere masses on a bounded hyperconvex domain always
admit global plurisubharmonic subextension with logarithmic growth at infinity
A priori -estimates for degenerate complex Monge-Amp\`ere equations
We study families of complex Monge-Amp\`ere equations, focusing on the case
where the cohomology classes degenerate to a non big class.
We establish uniform a priori -estimates for the normalized
solutions, generalizing the recent work of S. Kolodziej and G. Tian. This has
interesting consequences in the study of the K\"ahler-Ricci flow.Comment: 6 page
Maximal subextensions of plurisubharmonic functions
In this paper we are concerned with the problem of local and global
subextensions of (quasi-)plurisubharmonic functions from a "regular" subdomain
of a compact K\"ahler manifold. We prove that a precise bound on the complex
Monge-Amp\`ere mass of the given function implies the existence of a
subextension to a bigger regular subdomain or to the whole compact manifold. In
some cases we show that the maximal subextension has a well defined complex
Monge-Amp\`ere measure and obtain precise estimates on this measure. Finally we
give an example of a plurisubharmonic function with a well defined
Monge-Amp\`ere measure and the right bound on its Monge-Amp\`ere mass on the
unit ball in \C^n for which the maximal subextension to the complex
projective space \mb P_n does not have a globally well defined complex
Monge-Amp\`ere measure
Plurisubharmonic functions with weak singularities
We study the complex Monge-Amp\`ere operator in bounded hyperconvex domains
of \C^n. We introduce a scale of classes of weakly singular plurisubharmonic
functions : these are functions of finite weighted Monge-Amp\`ere energy. They
generalize the classes introduced by U.Cegrell, and give a stratification of
the space of (almost) all unbounded plurisubharmonic functions. We give an
interpretation of these classes in terms of the speed of decreasing of the
Monge-Amp\`ere capacity of sublevel sets and solve associated complex
Monge-Amp\`ere equations.Comment: 15 pages, dedicated to Christer Kiselman on the occasion of his
retiremen
Pluricomplex Green and Lempert functions for equally weighted poles
For a domain in , the pluricomplex Green function with
poles is defined as .
When there is only one pole, or two poles in the unit ball, it turns out to be
equal to the Lempert function defined from analytic disks into by . It is known
that we always have . In the more general case where we
allow weighted poles, there is a counterexample to equality due to Carlehed and
Wiegerinck, with equal to the bidisk.
Here we exhibit a counterexample using only four distinct equally weighted
poles in the bidisk. In order to do so, we first define a more general notion
of Lempert function "with multiplicities", analogous to the generalized Green
functions of Lelong and Rashkovskii, then we show how in some examples this can
be realized as a limit of regular Lempert functions when the poles tend to each
other. Finally, from an example where in the case of
multiple poles, we deduce that distinct (but close enough) equally weighted
poles will provide an example of the same inequality. Open questions are
pointed out about the limits of Green and Lempert functions when poles tend to
each other.Comment: 25 page
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