Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userParameter dependence of complex geodesics and its applications
73
0
0.0
(
0
)
Authors
Xieping Wang
Ask ChatGPT about the research
No Arabic abstract
We study the parameter dependence of complex geodesics with prescribed boundary value and direction on bounded strongly linearly convex domains.As an important application, we present a quantitative relationship between the regularity of the pluricomplex Poisson kernel of such a domain, which is a solution to a homogeneous complex Monge-Amp`{e}re equation with boundary singularity, and that of the boundary of the domain. Our results improve considerably previous ones in this direction due to Chang-Hu-Lee and Bracci-Patrizio.
rate research
Read More
We study complex geodesics and complex Monge-Amp`{e}re equations on bounded strongly linearly convex domains in $mathbb C^n$. More specifically, we prove the uniqueness of complex geodesics with prescribed boundary value and direction in such a domain, when its boundary is of minimal regularity. The existence of such complex geodesics was proved by the first author in the early 1990s, but the uniqueness was left open. Based on the existence and the uniqueness proved here, as well as other previously obtained results, we solve a homogeneous complex Monge-Amp`{e}re equation with prescribed boundary singularity, which was first considered by Bracci et al. on smoothly bounded strongly convex domains in $mathbb C^n$.
In this paper we study the following slice rigidity property: given two Kobayashi complete hyperbolic manifolds $M, N$ and a collection of complex geodesics $mathcal F$ of $M$, when is it true that every holomorphic map $F:Mto N$ which maps isometrically every complex geodesic of $mathcal F$ onto a complex geodesic of $N$ is a biholomorphism? Among other things, we prove that this is the case if $M, N$ are smooth bounded strictly (linearly) convex domains, every element of $mathcal F$ contains a given point of $overline{M}$ and $mathcal F$ spans all of $M$. More general results are provided in dimension $2$ and for the unit ball.
For a given ring (domain) in $overline{mathbb{R}}^n$ we discuss whether its boundary components can be separated by an annular ring with modulus nearly equal to that of the given ring. In particular, we show that, for all $nge 3,,$ the standard definition of uniformly perfect sets in terms of Euclidean metric is equivalent to the boundedness of moduli of separating rings. We also establish separation theorems for a half of a ring. As applications of those results, we will prove boundary Holder continuity of quasiconformal mappings of the ball or the half space in $mathbb{R}^n.$
Let $X$ be a compact Kahler manifold of dimension $n$ and $omega$ a Kahler form on $X$. We consider the complex Monge-Amp`ere equation $(dd^c u+omega)^n=mu$, where $mu$ is a given positive measure on $X$ of suitable mass and $u$ is an $omega$-plurisubharmonic function. We show that the equation admits a Holder continuous solution {it if and only if} the measure $mu$, seen as a functional on a complex Sobolev space $W^*(X)$, is Holder continuous. A similar result is also obtained for the complex Monge-Amp`ere equations on domains of $mathbb{C}^n$.
Let $Omega Subset mathbb C^n$ be a bounded strongly $m$-pseudoconvex domain ($1leq mleq n$) and $mu$ a positive Borel measure on $Omega$. We study the complex Hessian equation $(dd^c u)^m wedge beta^{n - m} = mu$ on $Omega$. First we give a sufficient condition on the measure $mu$ in terms of its domination by the $m$-Hessian capacity which guarantees the existence of a continuous solution to the associated Dirichlet problem with a continuous boundary datum. As an application, we prove that if the equation has a continuous $m$-subharmonic subsolution whose modulus of continuity satisfies a Dini type condition, then the equation has a continuous solution with an arbitrary continuous boundary datum. Moreover when the measure has a finite mass, we give a precise quantitative estimate on the modulus of continuity of the solution. One of the main steps in the proofs is to establish a new capacity estimate showing that the $m$-Hessian measure of a continuous $m$-subharmonic function on $Omega$ with zero boundary values is dominated by an explicit function of the $m$-Hessian capacity with respect to $Omega$, involving the modulus of continuity of $varphi$. Another important ingredient is a new weak stability estimate on the Hessian measure of a continuous $m$-subharmonic function.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
