Solving the d- and $\overline\partial$-equations in thin tubes and applications to mappings

Abstract

We construct a family of integral kernels for solving the ∂-equation with C k and Hölder estimates in thin tubes around totally real submanifolds in C n (theorems 1.1 and 3.1). Combining this with the proof of a theorem of Serre we solve the d-equation with estimates for holomorphic forms in such tubes (theorem 5.1). We apply these techniques and a method of Moser to approximate C k -diffeomorphisms between totally real submanifolds in C n in the C k -topology by biholomorphic mappings in tubes, by unimodular and symplectic biholomorphic mappings, and by automorphisms of C n .

Carleman approximation by holomorphic automorphisms of ℂ n

“…Our main Theorem 1.1 generalizes work of Forstnerič-Rosay [5], Forstnerič [3] and Forstnerič-Løw-Øvrelid [4], where similar results were proved for compact totally real manifolds. The proof of the main theorem depends on the Andersén-Lempert theory and also results on Carleman approximation by entire functions.…”

A Strong Oka Principle for Embeddings of Some Planar Domains into ℂ×ℂ∗

“…The result follows from their generalisation (also in [13,14]) of the Andersén-Lempert theorem, first proved in a different form by Andersén and Lempert [4]. These two results from [13,14], together with stronger results by Forstnerič and Løw [11], and Forstnerič, Løw and Øvrelid [12], are collectively known as Andersén-Lempert theorems for C n . Theorem 2.…”

“…In the same papers Forstnerič and Rosay also proved a more general result on approximation by automorphisms in a neighbourhood of a polynomially convex compact set K ⊂ C n . These results, together with stronger theorems by Forstnerič and Løw [11], and Forstnerič, Løw and Øvrelid [12], are referred to as Andersén-Lempert theorems for C n .…”

A soft Oka principle for proper holomorphic embeddings of open Riemann surfaces into (ℂ*)2

“…We now wish to apply Theorem 1.7 from [8] (see also the remark following that theorem regarding totally real submanifolds with boundary). Thinking of ψ t as a family of maps into (C * ) 2 , and then noting that ψ * t ω = 0 for all t, we see that ψ t is a totally real ω-flow of class C ∞ in the terminology of [8].…”

“…We now wish to apply Theorem 1.7 from [8] (see also the remark following that theorem regarding totally real submanifolds with boundary). Thinking of ψ t as a family of maps into (C * ) 2 , and then noting that ψ * t ω = 0 for all t, we see that ψ t is a totally real ω-flow of class C ∞ in the terminology of [8]. Although the result [8, Theorem 1.7] is given for ω the standard volume form on C n , its proof only requires that ω be closed (see also the remark preceding the statement of Theorem 1.7 in [8]) and that ω induces via contraction a non-degenerate pairing between holomorphic (n − 1)-forms and holomorphic vector fields.…”