Surfaces in self-dual Einstein manifolds and their twistor lifts

Abstract

The index of a compact orientable superminimal surface of a self-dual Einstein four-manifold M with positive scalar curvature is computed in terms of its genus and area. Also a lower bound of its nullity is obtained. Applications to the cases M = S 4 and M = CP 2 are given, characterizing the standard Veronese immersions and their twistor deformations as those with lowest index.