Solving the $d$ and $\bar\partial$-equation in thin tubes and applications to mappings

Details Solving the $d$ and $\bar\partial$-equation in thin tubes and applications to mappings Solving the $d$ and $\bar\partial$-equation in thin tubes and applications to mappings

2000-03-24

arxiv.org/abs/math/0003149v2

Michigan Math. J., 49 (2001), 369--416

Mathematics

Complex Variables

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Scientific paper

We construct a family of integral kernels for solving the \bar\partial equation with C^k and Holder estimates in thin tubes around totally real submanifolds in complex Eulidean spaces (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 holomorphic automorphisms of C^n.

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