Comparison of invariant metrics and distances on strongly pseudoconvex domains and worm domains

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作者
Filippo Bracci
John Erik Fornæss
Erlend Fornæss Wold
机构
[1] Università di Roma “Tor Vergata”,Dipartimento di Matematica
[2] Norwegian University of Science and Technology,Department of Mathematical Sciences
[3] University of Oslo,Department of Mathematics
来源
Mathematische Zeitschrift | 2019年 / 292卷
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摘要
We prove that for a strongly pseudoconvex domain D⊂Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D\subset \mathbb {C}^n$$\end{document}, the infinitesimal Carathéodory metric gC(z,v)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_C(z,v)$$\end{document} and the infinitesimal Kobayashi metric gK(z,v)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_K(z,v)$$\end{document} coincide if z is sufficiently close to bD and if v is sufficiently close to being tangential to bD. Also, we show that every two close points of D sufficiently close to the boundary and whose difference is almost tangential to bD can be joined by a (unique up to reparameterization) complex geodesic of D which is also a holomorphic retract of D. The same continues to hold if D is a worm domain, as long as the points are sufficiently close to a strongly pseudoconvex boundary point. We also show that a strongly pseudoconvex boundary point of a worm domain can be globally exposed; this has consequences for the behavior of the squeezing function.
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页码:879 / 893
页数:14
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