Biholomorphic mappings between weakly pseudoconvex domains

John Erik Fornaess

Vol. 74 (1978), No. 1, 63–65

DOI: 10.2140/pjm.1978.74.63

Abstract

Assume we have a biholomorphic mapping between weakly pseudoconvex domains. It is an old question whether this extends to a diffeomorphism between their closures. The well known theorem of Fefferman states that this is true for strongly pseudoconvex domains. We will show that if the map has a smooth extension to the boundary, then it cannot map an analytic disc in the boundary to a single point.

This is an immediate consequence of the following theorem.

Theorem. Assume Ω, W are bounded pseudoconvex sets with 𝒞² boundary in Cⁿ, and assume Φ : Ω → W is a biholomorphic map with a 𝒞²-extension Φ : Ω →W. Then Φ is a 𝒞²-diffeomorphism between Ω and W.

Mathematical Subject Classification 2000

Primary: 32F15

Secondary: 32H99

Milestones

Received: 12 May 1977

Published: 1 January 1978