Approximation of biholomorphic mappings by automorphisms of \(\mathbb{C}^ n\)
Let \(\Omega\) be open in \(\mathbb{C}^ n\) and \(\varphi_ t:\Omega \to\varphi_ t (\Omega) \subset \mathbb{C}^ n\), \(0 \leq t \leq 1\), a smooth family of biholomorhic mappings. The authors discuss at first the problem of sufficient (and necessary) conditions for an approximation of these mappings (uniformly on compact subsets of \(\Omega)\) by holomorphic automorphisms of \(\mathbb{C}^ n\). They recall (a generalized version of) a theorem of \textit{E. Andersén} and \textit{L. Lempert} [Invent. Math. 110, 371-388 (1992; Zbl 0770.32015)]: If every \(\varphi_ t (\Omega)\) is Runge in \(\mathbb{C}^ n\), then \(\varphi_ t\) can be approximated for every \(t \in[0,1]\) if and only if \(\varphi_ 0\) can be approximated. Conversely, if \(\Omega\) is a pseudoconvex Runge domain in \(\mathbb{C}^ n\) and the biholomorphic map \(\varphi:\Omega \to \varphi(\Omega) \subset \mathbb{C}^ n\) can be approximated by automorphisms of \(\mathbb{C}^ n\), then every compact subset \(K \subset \Omega\) has a neighborhood \(D\) and a smooth family \(\varphi_ t:D \to \varphi_ t (D) \subset \mathbb{C}^ n\) of biholomorphic mappings with \(\varphi_ 0=\text{id}_ D\), \(\varphi_ 1=\varphi\) and \(\varphi_ t(D)\) Runge in \(\mathbb{C}^ n\) for \(0 \leq t \leq 1\) (Theorem 1.1.). Using the fact that every polynomially convex subset \(K\) of \(\Omega\) has a basis of Stein neighborhoods that are Runge in \(\mathbb{C}^ n\), the authors show that for \(K\) compact and \(\varphi_ 0=\text{id}\) the image \(\varphi_ t (K)\) is polynomially convex, \(0 \leq t \leq 1\), if and only if \(K\) has a neighborhood \(U \subset \Omega\) with \(\varphi_ t (U)\) Runge in \(\mathbb{C}^ n\) for \(t \in[0,1]\) (Theorem 2.1). Now take two real-analytic submanifolds \(M_ 0\) and \(M_ 1\) of \(\mathbb{C}^ n\) and assume that they are diffeomorphic. When are \(M_ 0\) and \(M_ 1\) \(\mathbb{C}^ n\)-equivalent, i.e. when exists a biholomorphic map \(\varphi:\Omega \to \mathbb{C}^ n\) of a neighborhood \(\Omega\) of \(M_ 0\) with \(\varphi (M_ 0)=M_ 1\) such that \(\varphi\) can be approximated by automorphisms of \(\mathbb{C}^ n\)? In Section 3 the answer is given for a totally real and polynomially convex compact submanifold \(M_ 0\) in \(\mathbb{C}^ n\): \(M_ 0\) and \(M_ 1\) are \(\mathbb{C}^ n\)-equivalent if and only if they are isotopic in \(\mathbb{C}^ n\) through a family of totally real and polynomially convex submanifolds \(M_ t\) in \(\mathbb{C}^ n\), \(0 \leq t \leq 1\) (Theorem 3.1). The authors give several applications of this theorem in Section 4. Two real-analytic embeddings \(f_ 0\), \(f_ 1:M \to \mathbb{C}^ n\) are called \(\mathbb{C}^ n\)-equivalent if there is a neighborhood \(U\) of \(f_ 0(M)\) and a biholomorphic map \(\psi:U \to \psi (U) \subset \mathbb{C}^ n\) with \(\psi \circ f_ 0=f_ 1\), where \(\psi\) can be uniformly approximated by automorphisms of \(\mathbb{C}^ n\). Assume now that \(M\) is a compact real- analytic surface. There is a following result: If \(f_ 0(M)\) and \(f_ 1(M)\) are totally real and polynomially convex in \(\mathbb{C}^ n (n \geq 3)\), then \(f_ 0\) and \(f_ 1\) are \(\mathbb{C}^ n\)-equivalent (Theorem 5.1). Using the ``Jet Transversality Theorem'' of Thom it is additionally shown that there is an isotopy of embeddings \(\tau_ t:M \to \mathbb{C}^ n\), \(0 \leq t \leq 1\), connecting \(f_ 0\) and \(f_ 1\) such that every \(\tau_ t(M)\) is totally real and polynomially convex in \(\mathbb{C}^ n\). The set of totally real embeddings \(f:M \to \mathbb{C}^ n\), \(n \geq 3\), where the image is polynomially convex is open and dense in the space of all embeddings of \(M\) into \(\mathbb{C}^ n\) with respect to the Withney \({\mathcal C}^ \infty\) topology (Theorem 5.2). Finally the authors give necessary and sufficient conditions in topological terms for the existence of an approximation of a biholomorphic mapping \(\varphi:\Omega \to \mathbb{C}^ n\) by automorphisms of \(\mathbb{C}^ n\) in a neighborhood of a real-analytic, totally real, polynomially convex submanifold \(M \subset \Omega\) (Theorem 6.2). They close their interesting paper with several examples and open problems.
scientific article; zbMATH DE number 447106
scientific article