Approximation by automorphisms on smooth submanifolds of \(\mathbb{C}^ n\). (English) Zbl 0821.32028
The following result is proved. Suppose \(M \subset \mathbb{C} C^ n (n \geq 2)\) is a compact, totally real, polynomially convex submanifold of class \({\mathcal C}^ p (1 \leq p < \infty)\) and \(F : M \to \mathbb{C} C^ n\) is a \({\mathcal C}^ p\) mapping. Then the following are equivalent.
(i) For every \(\varepsilon > 0\) there exists a \(\Phi \in \operatorname{Aut} \mathbb{C} C^ n\) such that \(\| F - \Phi |_ -M \|_{{\mathcal C}^ p (M)} < \varepsilon\).
(ii) For every \(\varepsilon > 0\) there exists a totally real, polynomially convex isotopy \(F_ t : M \to \mathbb{C} C^ n (t \in [0,1])\) of class \({\mathcal C}^ p\) such that \(F_ 0\) is the identity on \(M\) and \(\| F_ 1 - F \|_{{\mathcal C}^ p (M)} < \varepsilon\).
This is then used to prove the following. Suppose \(M \subset \mathbb{C} C^ n\) is a compact, totally real, polynomial convex submanifold of dimension at most \(2n/3\) and of class \({\mathcal C}^ p\), \(2 \leq p < \infty\). Then for every \({\mathcal C}^ p\) mapping \(F : M \to \mathbb{C} C^ n\) and for every \(\varepsilon > 0\) there exists a holomorphic automorphism \(\Phi\) of \(\mathbb{C} C^ n\) such that \(\| F - \Phi |_ M \|_{{\mathcal C}^ p (M)} < \varepsilon\). If, in addition, \(F\) and \(M\) are real analytic and \(F(M)\) is totally real and polynomial convex, then \(F\) extends to a biholomorphic mapping in some neighborhood \(U\) of \(M\) which can be approximated uniformly on \(U\) by holomorphic automorphisms of \(\mathbb{C} C^ n\).
(i) For every \(\varepsilon > 0\) there exists a \(\Phi \in \operatorname{Aut} \mathbb{C} C^ n\) such that \(\| F - \Phi |_ -M \|_{{\mathcal C}^ p (M)} < \varepsilon\).
(ii) For every \(\varepsilon > 0\) there exists a totally real, polynomially convex isotopy \(F_ t : M \to \mathbb{C} C^ n (t \in [0,1])\) of class \({\mathcal C}^ p\) such that \(F_ 0\) is the identity on \(M\) and \(\| F_ 1 - F \|_{{\mathcal C}^ p (M)} < \varepsilon\).
This is then used to prove the following. Suppose \(M \subset \mathbb{C} C^ n\) is a compact, totally real, polynomial convex submanifold of dimension at most \(2n/3\) and of class \({\mathcal C}^ p\), \(2 \leq p < \infty\). Then for every \({\mathcal C}^ p\) mapping \(F : M \to \mathbb{C} C^ n\) and for every \(\varepsilon > 0\) there exists a holomorphic automorphism \(\Phi\) of \(\mathbb{C} C^ n\) such that \(\| F - \Phi |_ M \|_{{\mathcal C}^ p (M)} < \varepsilon\). If, in addition, \(F\) and \(M\) are real analytic and \(F(M)\) is totally real and polynomial convex, then \(F\) extends to a biholomorphic mapping in some neighborhood \(U\) of \(M\) which can be approximated uniformly on \(U\) by holomorphic automorphisms of \(\mathbb{C} C^ n\).
Reviewer: B.Gilligan (Regina)
MSC:
| 32M05 | Complex Lie groups, group actions on complex spaces |
| 32E30 | Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs |
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