Twistor geometry and warped product orthogonal complex structures

Abstract

The twistor space of the sphere S2n is an isotropic Grassmannian that fibers over S2n. An orthogonal complex structure (OCS) on a subdomain of S2n (a complex structure compatible with the round metric) determines a section of this fibration with holomorphic image. In this article, we use this correspondence to prove that any finite energy OCS on R6⊂S6 must be of a special warped product form, and we also prove that any OCS on R2n that is asymptotically constant must itself be constant. We give examples defined on R2n which have infinite energy and examples of nonstandard OCSs on flat tori in complex dimension 3 and greater.