On fixed points and determining sets for holomorphic automorphisms (Q1396332)

It is a result of classical function theory that if \(f: U\to U\) is a conformal self-mapping of a plane domain that fixes three distinct points then \(f(\zeta) \equiv\zeta\). The purpose of the present paper is to put this result into a geometrically natural context and to extend it to higher-dimensional domains and manifolds. In an attempt to extend this result to higher dimensions, one can ask the following question. For \(n\geq 2\), does there exist a positive integer \(k\) such that, if \(S\) is a set of \(k\) points in `general' position in \(\mathbb C^n\) and if \(D\subset\mathbb C^n\) is a domain containing \(S\), then each automorphism of \(D\) fixing \(S\) is necessarily the identity? The answer to that question is negative: no such `general' position can be defined to obtain a positive answer, as shown by the following theorem: For each finite set \(K=\{p_1,\dots,p_k\}\subset\mathbb C^n (n > 1)\), there exist a bounded domain \(D\) containing \(K\) and a subgroup \(H\subset\text{Aut}(D)\) isomorphic to \(U(n-1)\) (the complex unitary group of \(\mathbb C^{n-1}\)) such that each element of \(H\) fixes each point of \(K\). Consider the group of automorphisms, \(\Aut(M)\), of a complex connected, complete Hermitian manifold \(M\) of dimension \(m=\dim_{\mathbb C}M\geq 1\) such that each automorphism in \(\Aut(M)\) is an isometry. Then there exists an open dense subset \(W\) of the \((m+1)\)-fold product \(M\times\dots\times M\) such that any automorphism \(f\) fixing \(p_0,\dots,p_m\) coincides with the identity map of \(M\) whenever \((p_0,\dots,p_m)\in W\).