Self-dual Einstein Spaces, Heavenly Metrics and Twistors

Understand

Four-dimensional quaternion-Kahler metrics, or equivalently self-dual Einstein spaces M, are known to be encoded locally into one real function h subject to Przanowski's Heavenly equation.

• We elucidate the relation between this description and the usual twistor description for quaternion-Kahler spaces.
• In particular, we show that the same space M can be described by infinitely many different solutions h, associated to different complex (local) submanifolds on the twistor space, and therefore to different (local) integrable complex structures on M.
• We also study quaternion-Kahler deformations of M and, in the special case where M has a Killing vector field, show that the corresponding variations of h are related to eigenmodes of the conformal Laplacian on M.