On the characterization of Grauert tubes covered by the ball (Q1365816)

A real analytic manifold \(X\) admits a complexification \(X_C\), a complex manifold which contains \(X\) as the real points, or fixed points of an antiholomorphic involution, or complex conjugation \(\sigma: X_C \to X_C\). If \(X\) is compact there is a neighborhood \(M\) of \(X\) in \(X_C\) and a smooth strictly plurisubharmonic function \(\rho: M\to [0,1)\) such that \(X\) is the zero set of \(\rho\). Under two extra data the function \(\rho\) is unique. Thus, the set \(X^r_C= \{\rho<r\}\) is called the Grauert tube of radius \(r\) over the center \(X= \{\rho=0\}\). The author proves the following Theorem: Let \(\Omega= X^r_C\) be a Grauert tube covered by the unit ball \(B^n\), then \(X= H^n/ \Gamma\) for some discrete subgroup \(\Gamma\) of \(O(n,1)\) and \(\Omega= B^n/ \Gamma\), where \(H^n =\{x\in R^n;\;| x |^2 <1\}\) is the real hyperbolic space with a certain complete Riemannian metric with constant sectional curvature.