## Comparison of invariant metrics and distances on strongly pseudoconvex domains and worm domains. (arXiv:1710.04192v1 [math.CV])

We prove that for a strongly pseudoconvex domain $D\subset\mathbb C^n$, the
infinitesimal Carath\'{e}odory 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.查看全文