Twistor geometry and warped product orthogonal complex structures

Abstract

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