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Orthogonal complex structures on domains in \({\mathbb {R}^4}\)

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Abstract

An orthogonal complex structure on a domain in \({\mathbb {R}^4}\) is a complex structure which is integrable and is compatible with the Euclidean metric. This gives rise to a first order system of partial differential equations which is conformally invariant. We prove two Liouville-type uniqueness theorems for solutions of this system, and use these to give an alternative proof of the classification of compact locally conformally flat Hermitian surfaces first proved by Pontecorvo. We also give a classification of non-degenerate quadrics in \({\mathbb {CP}^3}\) under the action of the conformal group SO °(1, 5). Using this classification, we show that generic quadrics give rise to orthogonal complex structures defined on the complement of unknotted solid tori which are smoothly embedded in \({\mathbb {R}^4}\) .

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Authors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Turin, Italy

    Simon Salamon

  2. Department of Mathematics, University of Wisconsin, Madison, WI, 53706, USA

    Jeff Viaclovsky

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  1. Simon Salamon
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  2. Jeff Viaclovsky
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Correspondence to Jeff Viaclovsky.

Additional information

To Nigel Hitchin on the occasion of his 60th birthday.

This work was partially supported by MIUR (Metriche Riemanniane e Varietà Differenziabili, PRIN05) and NSF Grant DMS-0503506.

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Salamon, S., Viaclovsky, J. Orthogonal complex structures on domains in \({\mathbb {R}^4}\) . Math. Ann. 343, 853–899 (2009). https://doi.org/10.1007/s00208-008-0293-5

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