Toric anti-self-dual Einstein metrics via complex geometry

Details Toric anti-self-dual Einstein metrics via complex geometry Toric anti-self-dual Einstein metrics via complex geometry

2006-09-18

arxiv.org/abs/math/0609487v2

Math. Ann. 340 (2008), no 1. 143--157

Mathematics

Differential Geometry

v2. Published version. Additional references. 14 pages

Scientific paper

Using the twistor correspondence, we give a classification of toric anti-self-dual Einstein metrics: each such metric is essentially determined by an odd holomorphic function. This explains how the Einstein metrics fit into the classification of general toric anti-self-dual metrics given in an earlier paper (math.DG/0602423). The results complement the work of Calderbank-Pedersen (math.DG/0105263), who describe where the Einstein metrics appear amongst the Joyce spaces, leading to a different classification. Taking the twistor transform of our result gives a new proof of their theorem.

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