7,557 research outputs found
Parameter dependence of the Bergman kernels
Let be a family of bounded pseudoconvex domains and
. Let denote the Bergman kernel with
weight on . We study the continuity and H\"older
continuity of in . Several applications to singularity theory of
psh functions are given, including a new proof of the openness theorem
Equivalence of the Bergman and Teichmuller metrics on Teichmuller spaces
We prove that the Bergman and the Teichmuller metrics are equivalent on
Teichmuller spaces.Comment: A serious mistake in the previous manuscript was correcte
Weighted Bergman spaces and the equation
We give a H\"ormander type estimate for the equation
with respect to the measure , , on any
bounded pseudoconvex domain with boundary. Several applications to the
function theory of weighed Bergman spaces are given,
including a corona type theorem, a Gleason type theorem, together with a
density theorem. We investigate in particular the boundary behavior of
functions in by proving an analogue of the Levi problem
for and giving an optimal Gehring type estimate for
functions in . A vanishing theorem for is
established for arbitrary bounded domains. Relations between the weighted
Bergman kernel and the Szeg\"o kernel are also discussed.Comment: 23 pages; Some minor mistakes are corrected; to appear in Trans. AM
Weighted Bergman kernel, directional Lelong number and John-Nirenberg exponent
Let be a plurisubharmonic function on the closed unit ball and
the Bergman kernel on the unit ball with respect to the weight
. We show that the boundary behavior of is determined by
certain directional Lelong number of for all smaller than the
John-Nirenberg exponent of associated to certain family of nonisotropic
balls, which is always positive
A simple proof of the Ohsawa-Takegoshi extension theorem
We give a direct proof of the Ohsawa-Takegoshi by solving directly the d-bar
equation
Hardy-Sobolev type inequalities and their applications
This paper is devoted to various applications of Hardy-Sobolev type
inequalities. We derive a new estimate for the equation
on which yields a quantitative generalization of the Hartogs
extension theorem to the case when the singularity set is not necessary
compact. We show that for any negative subharmonic function on , , the BMO norm of is bounded above by
and satisfies a reverse H\"older inequality for every
. We also show that every plurisubharmonic function is locally BMO.
Several Liouville theorems for subharmonic functions on complete Riemannian
manifolds are given. As a consequence, we get a Margulis type theorem that if a
bounded domain in covers a Zariski open set in a projective
algebraic variety, then the group of deck transformations of the covering has
trivial center.Comment: A proof of the Hartogs extension theorem for
pluriharmonic functions is adde
Convergence of Riemannian surfaces and convergence of the Bergman kernel
Let be a sequence of complete Riemannian surfaces which converges
in the sense of Cheeger-Gromov to a complete Riemannian surface . We study
the convergence of the Bergman kernel of by using isoperimetric
inequalities
General rogue waves and their dynamics in several reverse time integrable nonlocal nonlinear equations
A study of general rogue waves in some integrable reverse time nonlocal
nonlinear equations is presented. Specifically, the reverse time nonlocal
nonlinear Schr\"odinger (NLS) and nonlocal Davey-Stewartson (DS) equations are
investigated, which are nonlocal reductions from the AKNS hierarchy. By using
Darboux transformation (DT) method, several types of rogue waves are
constructed. Especially, a unified binary DT is found for this nonlocal DS
system, thus the solution formulas for nonlocal DSI and DSII equation can be
written in an uniform expression. Dynamics of these rogue waves is separately
explored. It is shown that the (1+1)-dimensional rogue waves in nonlocal NLS
equation can be bounded for both x and t, or develop collapsing singularities.
It is also shown that the (1+2)-dimensional line rogue waves in the nonlocal DS
equations can be bounded for all space and time, or have finite-time
blowing-ups. All these types depend on the values of free parameters introduced
in the solution. In addition, the dynamics patterns in the multi- and
higher-order rogue waves exhibits more richer structures, most of which have no
counterparts in the corresponding local nonlinear equations.Comment: 22 pages, 12 figure
Dynamics of Rogue Waves in the Partially PT-symmetric Nonlocal Davey-Stewartson Systems
In this work, we study the dynamics of rogue waves in the partially
-symmetric nonlocal Davey-Stewartson(DS) systems. Using the Darboux
transformation method, general rogue waves in the partially
-symmetric nonlocal DS equations are derived. For the partially
-symmetric nonlocal DS-I equation, the solutions are obtained and
expressed in term of determinants. For the partially -symmetric DS-II
equation, the solutions are represented as quasi-Gram determinants. It is shown
that the fundamental rogue waves in these two systems are rational solutions
which arises from a constant background at , and develops
finite-time singularity on an entire hyperbola in the spatial plane at the
critical time. It is also shown that the interaction of several fundamental
rogue waves is described by the multi rogue waves. And the interaction of
fundamental rogue waves with dark and anti-dark rational travelling waves
generates the novel hybrid-pattern waves. However, no high-order rogue waves
are found in this partially -symmetric nonlocal DS systems. Instead,
it can produce some high-order travelling waves from the high-order rational
solutions.Comment: 22 pages, 26 figure
Stability of the Bergman kernel on a tower of coverings
We obtain several results about stability of the Bergman kernel on a tower of
coverings on complex manifolds. An effective version of Rhodes' result is given
for a tower of coverings on a compact Riemann surface of genus greater than or
equal to 2. Stability of the Bergman kernel is established for towers of
coverings on hyperbolic Riemann surfaces and on complete Kaehler manifolds
satisfying certain potential conditions. As a consequence, stability of the
Bergman kernel is established for any tower of coverings of Riemann surfaces
when the top manifold is simply-connected.Comment: 22 page
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