Moduli Space of Branched Superminimal Immersions of a Compact Riemann Surface Into

In this paper we describe the moduli spaces of degree d branched superminimal immersions of a compact Riemann surface of genus g into S4. We prove that when d ≥ max{2g, g + 2}, such spaces have the structure of projectivized fibre products and are path-connected quasi-projective varieties of dimension 2d − g + 4. This generalizes known results for spaces of harmonic 2-spheres in S4. In the Calabi construction (see [C]) the space of harmonic maps (or, equivalently, the space of branched minimal immersions) of S into S decomposes into a union of moduli spaces Hk labelled by the harmonic degree k ≥ 0. (This means that if f ∈ Hk, then Area(f(S)) = Energy(f) = 4πk.) When f is linearly full (that is, its image is not contained in a totally geodesic subsphere), then either f or −f has a holomorphic horizontal lift to the twistor space of S such that its image is a curve of degree k. In the case when n = 2, if f is totally geodesic, then both f and −f have holomorphic horizontal lifts to the twistor space CP, and their images in CP are linear projective lines. The study of the space of harmonic maps of S into S (modulo the antipodal map on S) thus reduces to the study of holomorphic horizontal curves (of genus 0) in CP. In [L] the study of holomorphic horizontal curves of degree k and genus 0 in CP was reduced to the study of the moduli spaceMk of pairs of meromorphic functions of degree k with the same ramification divisor. Recall from Wirtinger’s inequality that a holomorphic curve in a Kähler manifold is area minimizing in its homology class. Since the twistor projection CP → S is a Riemannian submersion, the projection of a holomorphic horizontal curve in CP is automatically minimal in S. By general twistor theory, such maps are also harmonic (see [BR] and [ES] for example). For a fixed compact Riemann surface Σ of genus g > 0, we shall consider the special class of harmonic maps from Σ to S which admit holomorphic horizontal liftings to CP. Such maps are called “branched superminimal immersions”. In 1982 Robert Bryant obtained the following result: Theorem ([Br]). Every compact Riemann surface admits a superminimal immersion into S. In this paper, we shall study the moduli spaces of all such maps. We shall give a direct construction for holomorphic horizontal curves of genus g. Since every such curve in CP has a degree, we can classify the moduli spaces by the degree of the 1991 Mathematics Subject Classification. Primary 58D27; Secondary 58E20, 53A10, 53C15, 32L25. *Submitted for publication.