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

2017/10/11 by Filippo Bracci, Bracci, Filippo, John Erik Fornæss +3
Mathematics · #Holomorphic and Operator Theory #Analytic and geometric function theory #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1710.04192

Abstract

We prove that for a strongly pseudoconvex domain D\⊂ mathbb Cⁿ, the\ninfinitesimal Carath 'eodory metric g_C(z,v) and the infinitesimal Kobayashi\nmetric g_K(z,v) coincide if z is sufficiently close to bD and if v is\nsufficiently close to being tangential to bD. Also, we show that every two\nclose points of D sufficiently close to the boundary and whose difference is\nalmost tangential to bD can be joined by a (unique up to reparameterization)\ncomplex geodesic of D which is also a holomorphic retract of D.\n The same continues to hold if D is a worm domain, as long as the points are\nsufficiently close to a strongly pseudoconvex boundary point. We also show that\na strongly pseudoconvex boundary point of a worm domain can be globally\nexposed, this has consequences for the behavior of the squeezing function.\n

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