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The Quarterly Journal of Mathematics Advance Access published online on April 19, 2008
The Quarterly Journal of Mathematics, doi:10.1093/qmath/han002
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© 2008. Published by Oxford University Press. All rights reserved For permissions, please email: journals.permissions@oxfordjournals.org

THE ADAPTED COMPLEXIFICATION OF THE TWO-SPHERE WITH A LIOUVILLE METRIC

Raúl M. Aguilar{dagger}

Department of Science and Mathematics, Massachusetts Maritime Academy, Buzzards Bay, Massachusetts, USA

{dagger} Email: raguilar{at}maritime.edu

Received 20 April 2007; revised 22 November 2007
  Abstract

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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