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arXiv · 1710.04192

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

Filippo Bracci·John Erik Fornaess·Erlend Fornaess Wold
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Abstract

We prove that for a strongly pseudoconvex domain $D\subset\mathbb C^n$, the infinitesimal Carathéodory metric $g_C(z,v)$ and the infinitesimal Kobayashi metric $g_K(z,v)$ 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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Filippo Bracci, John Erik Fornaess, Erlend Fornaess Wold. 2018-06-03. Comparison of invariant metrics and distances on strongly pseudoconvex domains and worm domains. https://arxiv.org/abs/1710.04192

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