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Register a new userQuantitative localization and comparison of invariant distances of domains in $mathbb C^n$
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We obtain explicit bounds on the difference between local and global Kobayashi distances in a domain of $mathbb C^n$ as the points go toward a boundary point with appropriate geometric properties. We use this for the global comparison of various invariant distances. We provide some sharp estimates in dimension $1$.
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Let $DsubsetC^n$ be a bounded, strongly pseudoconvex domain whose boundary $bD$ satisfies the minimal regularity condition of class $C^2$. A 2017 result of Lanzani & Stein states that the Cauchy--SzegH{o} projection $EuScript S_omega$ defined with respect to any textit{Leray Levi-like} measure $omega$ is bounded in $L^p(bD, omega)$ for any $1<p<infty$. (For this class of domains, induced Lebesgue measure $sigma$ is Leray Levi-like.) Here we show that $EuScript S_omega$ is in fact bounded in $L^p(bD, Omega_p)$ for any $1<p<infty$ and for any $Op$ in the far larger class of textit{$A_p$-like} measures (modeled after the Muckenhoupt $A_p$-weights for $sigma$). As an application, we characterize boundedness and compactness in $L^p(bD, Omega_p)$ for $1<p<infty$, of the commutator $[b, EuScript S_omega]$. We next introduce the holomorphic Hardy spaces $H^p(bD, Omega_p)$, $1<p<infty$, and we characterize boundedness and compactness in $L^2(bD, Omega_2)$ of the commutator $displaystyle{[b,EuScript S_{Omega_2}]}$ of the Cauchy--SzegH{o} projection defined with respect to any $A_2$-like measure $Omega_2$. Earlier closely related results rely upon an asymptotic expansion, and subsequent pointwise estimates, of the Cauchy--SzegH o kernel that are not available in the settings of minimal regularity {of $bD$} and/or $A_p$-like measures.
We give a complete characterization of those disk bundles over surfaces which embed as rationally convex strictly pseudoconvex domains in $mathbb{C}^2$. We recall some classical obstructions and prove some deeper ones related to symplectic and contact topology. We explain the close connection to Lagrangian surfaces with isolated singularities and develop techniques for constructing such surfaces. Our proof also gives a complete characterization of Lagrangian surfaces with open Whitney umbrellas, answering a question first posed by Givental in 1986.
We study the homeomorphic extension of biholomorphisms between convex domains in $mathbb C^d$ without boundary regularity and boundedness assumptions. Our approach relies on methods from coarse geometry, namely the correspondence between the Gromov boundary and the topological boundaries of the domains and the dynamical properties of commuting 1-Lipschitz maps in Gromov hyperbolic spaces. This approach not only allows us to prove extensions for biholomorphisms, but for more general quasi-isometries between the domains endowed with their Kobayashi distances.
The Leray transform and related boundary operators are studied for a class of convex Reinhardt domains in $mathbb C^2$. Our class is self-dual; it contains some domains with less than $C^2$-smooth boundary and also some domains with smooth boundary and degenerate Levi form. $L^2$-regularity is proved, and essential spectra are computed with respect to a family of boundary measures which includes surface measure. A duality principle is established providing explicit unitary equivalence between operators on domains in our class and operators on the corresponding polar domains. Many of these results are new even for the classical case of smoothly bounded strongly convex Reinhardt domains.
These are notes for the CIME school on Complex non-Kahler geometry from July 9th to July 13th of 2018 in Cetraro, Italy. It is an overview of different properties of a class of non-Kahler compact complex manifolds called LVMB manifolds, obtained as the Hausdorff space of leaves of systems of commuting complex linear equations in an open set in complex projective space ${{mathbb P}_{mathbb C}}^{n-1}$
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