Qs. Chi, The dimension of the moduli space of superminimal surfaces of a fixed degree and conformal structure in the 4-sphere, TOHOKU MATH, 52(2), 2000, pp. 299-308
It was established by X. Mo and the author that the dimension of each irred
ucible component of the moduli space M-d,M-g(X) of branched superminimal im
mersions of degree d from a Riemann surface X of genus g into CP3 lay betwe
en 2d-4g+4 and 2d-g+4 ford sufficiently large, where the upper bound was al
ways assumed by the irreducible component of totally geodesic branched supe
rminimal immersions and the lower bound was assumed by all nontotally geode
sic irreducible components of M-6,M-1(T) for any torus T. It is shown, via
deformation theory, in this note that for d = 89 + 1 + 3k, k greater than o
r equal to 0, and any Riemann surface X of g greater than or equal to 1, th
e above lower bound is assumed by at least one irreducible component of Md,
g(X).