## Tohoku Mathematical Journal 2000 June SECOND SERIES VOL. 52, NO. 2

 Tohoku Math. J. 52 (2000), 299-308

Title THE DIMENSION OF THE MODULI SPACE OF SUPERMINIMAL SURFACES OF A FIXED DEGREE AND CONFORMAL STRUCTURE IN THE 4-SPHERE

Author Quo-Shin Chi

Abstract. It was established by X. Mo and the author that the dimension of each irreducible component of the moduli space $\mathcal{M}_{d,g}(X)$ of branched superminimal immersions of degree $d$ from a Riemann surface $X$ of genus $g$ into $\boldsymbol{C}P^3$ 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 {\it totally geodesic} branched superminimal immersions and the lower bound was assumed by all {\it nontotally geodesic} irreducible components of $\mathcal{M}_{6,1}(T)$ for any torus $T$. It is shown, via deformation theory, in this note that for $d=8g+1+3k$, $k\geq 0$, and any Riemann surface $X$ of $g\geq 1$, the above lower bound is assumed by at least one irreducible component of $\mathcal{M}_{d,g}(X)$.