Orthogonal complex structures on domains in R^4

Details Orthogonal complex structures on domains in R^4 Orthogonal complex structures on domains in R^4

2007-04-25

arxiv.org/abs/0704.3422v2

Final version: Math. Annalen 343, no. 4, 2009, 853-899

Mathematics

Differential Geometry

42 pages. Version 2 contains several improvements and simplifications throughout. Material from the first version on more gene

Scientific paper

An orthogonal complex structure on a domain in 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 CP^3 under the action of the conformal group. 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 R^4.

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