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arXiv · 0905.3662

Twistor geometry and warped product orthogonal complex structures

Lev Borisov·Simon Salamon·Jeff Viaclovsky
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Abstract

The twistor space of the sphere S^{2n} is an isotropic Grassmannian that fibers over S^{2n}. An orthogonal complex structure 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 paper, we use this correspondence to prove that any finite energy orthogonal complex structure on R^6 must be of a special warped product form, and we also prove that any orthogonal complex structure on R^{2n} that is asymptotically constant must itself be constant. We will also give examples defined on R^{2n} which have infinite energy, and examples of non-standard orthogonal complex structures on flat tori in complex dimension three and greater.

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Lev Borisov, Simon Salamon, Jeff Viaclovsky. 2009-05-22. Twistor geometry and warped product orthogonal complex structures. https://doi.org/10.1215/00127094-2010-068

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