Geodesic
Complex Structures and Conformal Geometry
Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 1, pp. 199-224

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Zbl MR
A characterization of certain complex structures on conformally-flat domains in real dimension 4 is carried out in the context of Hermitian geometry and twistor spaces. The presentation is motivated by some classical surface theory, whilst the problem itself leads to a refined classification of quadrics in complex projective 3-space. The main results are sandwiched between general facts in real dimension 2n and some concluding examples in real dimension 6. 
Salamon, Simon. Complex Structures and Conformal Geometry. Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 1, pp. 199-224. http://geodesic.mathdoc.fr/item/BUMI_2009_9_2_1_a9/
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     title = {Complex {Structures} and {Conformal} {Geometry}},
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     pages = {199--224},
     year = {2009},
     volume = {Ser. 9, 2},
     number = {1},
     zbl = {1182.53043},
     mrnumber = {2493651},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2009_9_2_1_a9/}
}
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