Parameter dependence of the Bergman kernels

Let $\{\Omega_t:-1<t<1\}$ be a family of bounded pseudoconvex domains and $\varphi_t\in PSH(\Omega_t)$. Let $K_t(z,w)$ denote the Bergman kernel with weight $\varphi_t$ on $\Omega_t$. We study the continuity and H\"older continuity of $K_t(z,w)$ in $t$. 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 ∂ˉ−\bar{\partial}-equation

We give a H\"ormander type $L^2-$estimate for the $\bar{\partial}-$equation with respect to the measure $\delta_\Omega^{-\alpha}dV$, $\alpha<1$, on any bounded pseudoconvex domain with $C^2-$boundary. Several applications to the function theory of weighed Bergman spaces $A^2_\alpha(\Omega)$ 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 $A^2_\alpha(\Omega)$ by proving an analogue of the Levi problem for $A^2_\alpha(\Omega)$ and giving an optimal Gehring type estimate for functions in $A^2_\alpha(\Omega)$. A vanishing theorem for $A^2_1(\Omega)$ 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 $\psi$ be a plurisubharmonic function on the closed unit ball and $K_{t\psi}(z)$ the Bergman kernel on the unit ball with respect to the weight $t\psi$. We show that the boundary behavior of $K_{t\psi}(z)$ is determined by certain directional Lelong number of $\psi$ for all $t$ smaller than the John-Nirenberg exponent of $\psi$ 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 $L^2$ estimate for the $\bar{\partial}-$equation on ${\mathbb C}^n$ 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 $\psi$ on ${\mathbb R}^n$, $n>2$, the BMO norm of $\log |\psi|$ is bounded above by $2\sqrt{n-2}$ and $|\psi|^\gamma$ satisfies a reverse H\"older inequality for every $0<\gamma<1$. 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 ${\mathbb C}^n$ covers a Zariski open set in a projective algebraic variety, then the group of deck transformations of the covering has trivial center.Comment: A $\partial\bar{\partial}-$proof of the Hartogs extension theorem for pluriharmonic functions is adde

Convergence of Riemannian surfaces and convergence of the Bergman kernel

Let $\{M_j\}$ be a sequence of complete Riemannian surfaces which converges in the sense of Cheeger-Gromov to a complete Riemannian surface $M$. We study the convergence of the Bergman kernel $K_{M_j}$ of $M_j$ 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 $\cal{PT}$-symmetric nonlocal Davey-Stewartson(DS) systems. Using the Darboux transformation method, general rogue waves in the partially $\cal{PT}$-symmetric nonlocal DS equations are derived. For the partially $\cal{PT}$-symmetric nonlocal DS-I equation, the solutions are obtained and expressed in term of determinants. For the partially $\cal{PT}$-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 $t\rightarrow -\infty$, 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 $\cal{PT}$-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