On rigidity of Grauert tubes over Riemannian manifolds of constant curvature
Let \((X,g)\) be a real analytic Riemannian manifold. Then in some neighborhood \(\Omega\) of \(X\) in \(TX\) there exists a unique complex structure such that \(\Omega\) is a complexification of \(X\), i.e. there is an anti-holomorphic involutoin of \(\Omega\) for which \(X\) is a fixed point set and such that the map \(f_\gamma(\sigma+ i\tau):= (\tau\gamma'(\sigma),\gamma(\sigma))\) from \(\mathbb{C}\) to \(TX\) is holomorphic for \(\sigma+ i\tau\in\mathbb{C}\) where \(f\) is defined. In this complex structure, called adapted, the function \(\rho(V)=\|V\|^2= g(V,V)\) is strictly plurisubharmonic. The Grauert tube of radius \(r\) over \((X,g)\) is \(X^r_{\mathbb{C}}= \{V\in TX:\|V\|< r\}\) equipped with the adapted structure. The maximal radius \(r_{\max}\) is the supremum of \(r\) such that \(X^r_{\mathbb{C}}\) exists as a complex manifold. \(X\) is called the center of \(X^r_{\mathbb{C}}\). By this construction one has the natural inclusion \(\text{Isom}(X)\to \text{Aut}(X^r_{\mathbb{C}})\). The main result of the paper is the following theorem: Let \(X\) be a compact real analytic Riemannian manifold of constant curvature. Then for each Grauert tube \(X^r_{\mathbb{C}}\) over \(X\) with \(r< r_{\max}\) the inclusion \(\text{Isom}(X)\to \text{Aut}(X^r_{\mathbb{C}})\) is an isomorphism.
scientific article; zbMATH DE number 1761100