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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Based on the introductory message on the board (I have not yet read the link to more) I find the geometrical considerations above of a high order but missing some essential worldview interpretations as to the actual possibilities of the geometry as information and complexity. In particular it does not make a clear statement about isotopy of radial centers in a general space as to there being alternative representations. There is more than the appling of FFT as if a modification of continuity- space is more general and abstract than that and we perceive things more from a unified continuous and descete place of a more unified physics and mathematics. But the transcendental is relavant provided we distinguish where we can these grounds of set theory.