The moduli space of branched superminimal surfaces of a fixed degree, genus and conformal structure in the four-sphere

Abstract

0. Introduction. Minimal immersions from a Riemann surface M into Sn were studied by Calabi ([3]) and Chern ([4]), among many authors. To such an immesion F in S4, they found a holomorphic quartic form QF (to be defined in Section 1) on M . A superminimal immersion is one for which QF = 0, which is always the case when M = S 2. In [2], Bryant studied a superminimal immersion of a higher genus into S4 by lifting it to CP 3, the twistor space of S4. The lift of a superminimal immersion is a holomorphic curve, of the same degree as that of the immersion, which is horizontal with respect to the twistorial fibration; more precisely, it is a holomorphic curve in CP 3 satisfying the differential equation