The Dimension of the Moduli Space of Superminimal Surfaces of a Fixed Degree and Conformal Structure in the 4-sphere

It was established by X. Mo and the author that the dimension of each irreducible component of the moduli space Λd^^giX) of branched superminimal immersions of degree d from a Riemann surface X of genus g into C P 3 lay between 2d—4g+4 and 2d — g+4 for d sufficiently large, where the upper bound was always assumed by the irreducible component of totally geodesic branched superminimal immersions and the lower bound was assumed by all nontotally geodesic irreducible components of M.^^\{T) for any torus T. It is shown, via deformation theory, in this note that for d = Sg + 1 + 3k, k > 0, and any Riemann surface X of g > 1, the above lower bound is assumed by at least one irreducible component of 0. The dimension and irreducibility are two fundamental questions when dealing with moduli spaces. In [2] Calabi studied minimal 2-spheres in an ambient round sphere, where he showed that the ambient sphere must be of even dimension if the minimal 2-sphere is linearly full in the ambient sphere. Moreover, all the minimal 2-spheres are obtained by projecting horizontal holomorphic rational curves from the twistor space of the ambient sphere S into S. Here, the twistor space of S is the Hermitian symmetric space of pointwise orthogonal complex structures of S, and horizontally refers to the horizontal distribution of the twistor space naturally induced by the Riemannian connection of S. In general, the projection of any horizontal holomorphic curve from the twistor space into S is a minimal surface called a (branched) superminimal surface. The twistor space of S happens to be the pleasant CP, where a horizontal holomorphic curve satisfies the differential equation (1) zodzi — z\dzo + zidz3 — z^dzi = 0 with the homogeneous coordinates [zo z\ : zi : Z3] of C P 3 , with respect to which Bryant [1] proved the existence of branched superminimal surfaces of arbitrary genus and conformal structure in S. Loo [10] and Verdier [12] later studied the moduli space of the branched superminimal spheres of a fixed area (equal to a constant multiple of the degree d of the corresponding horizontal holomorphic curves). Subsequently Mo and I [3] investigated the moduli space Md,g(X) of branched superminimal surfaces of a fixed degree d from any Riemann surface X of genus g into the four-sphere. By definition Md,g(X) is the variety of all horizontal holomorphic maps from X into CP satisfying (1). From this equation one 1991 Mathematics Subject Classification. Primary 53C42.