Abstract
The twistor space of the sphere S 2 n is an isotropic Grassmannian that fibers over S 2 n . An orthogonal complex structure (OCS) on a subdomain of S 2 n (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 R 6 ⊂ S 6 must be of a special warped product form, and we also prove that any OCS on R 2 n that is asymptotically constant must itself be constant. We give examples defined on R 2 n which have infinite energy and examples of nonstandard OCSs on flat tori in complex dimension 3 and greater.