On the rigidity of non-positively curved Grauert tubes
Let \(X\) be a compact Riemannian manifold, \(\mathbb CX\) its complexification. \textit{V. Guillemin} and \textit{M. Stenzel} [J. Differ. Geom. 34, 561-570 (1991; Zbl 0746.32005)], and also \textit{L. Lempert} and \textit{R. Szöke} [Math. Ann. 290, 689-712 (1991; Zbl 0752.32008)] proved the existence and uniqueness of a non-negative strictly plurisubharmonic function \(\rho \) in a neighbourhood \(U\) of \(X\) in \(\mathbb CX\) with \(X=\rho ^{-1}(0)\), such that \(\sqrt{\rho }\) satisfies the complex Monge-Ampère equation on \(U\setminus X\), and the restriction to \(X\) of the Kähler metric induced from the Kähler form \(\frac{i}2\partial \overline{\partial }\rho \) coincides with the original Riemannian metric on \(X\). The set \(X_{\mathbb C}^r=\{ \rho <r^2\} \) is called the Grauert tube of radius \(r\) over the center \(X\). The author finds conditions under which the equality \(X_{\mathbb C}^r=Y_{\mathbb C}^r\) for two manifolds \(X,Y\) implies the equality \(X=Y\). The conditions are formulated in terms of the curvature with respect to some Kähler metric on \(X_{\mathbb C}^r\). Several examples are given.
scientific article; zbMATH DE number 1268071