Second variation of superminimal surfaces into self-dual Einstein four-manifolds

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 = CP2 are given, characterizing the standard Veronese immersions and their twister deformations as those with lowest index.