Vol. 74, No. 1, 1978

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Biholomorphic mappings between weakly pseudoconvex domains

John Erik Fornaess

Vol. 74 (1978), No. 1, 63–65

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
Received: 12 May 1977
Published: 1 January 1978
John Erik Fornaess