We show that the two-sphere with a Riemannian metric that is Liouville with finite isometry group does not admit an unbounded adapted complexification in the sense of Lempert and Szoke and of Guillemin and Stenzel; that is, its Grauert tube cannot have infinite radius. We prove this by first extending a classical theorem valid for umbilical geodesics in a triaxial ellipsoid to general Liouville metrics. Furthermore, we derive an isometric rigidity result for the Monge–Ampère foliation of a two-dimensional Grauert tube with infinite radius.



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