Orthogonal complex structures on domains in mathbb R4{mathbb {R}^4}

An orthogonal complex structure on a domain in 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 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 . 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.