Abstract
We show that an n-dimensional real ellipsoid in ℝn+1 with the induced Riemannian metric does not admit an unbounded adapted complexification in the sense of Lempert/Szőke and Guillemin/Stenzel, unless it is a round sphere. In other words, an ellipsoid whose (maximal) Grauert tube has infinite radius must be a round sphere. For the proof we take advantage of the integrability of the geodesic flow and use a classical theorem on umbilic geodesics. We carry out an extension of this result to Liouville metrics elsewhere.
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