Tohoku Mathematical Journal

The dimension of the moduli space of superminimal surfaces of a fixed degree and conformal structure in the 4-sphere

Quo-Shin Chi

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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 $C P^3$ 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 $\mathcal{M}_{d,g}(X)$.

Article information

Tohoku Math. J. (2), Volume 52, Number 2 (2000), 299-308.

First available in Project Euclid: 3 May 2007

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Zentralblatt MATH identifier

Primary: 53C43: Differential geometric aspects of harmonic maps [See also 58E20]
Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42] 58D27: Moduli problems for differential geometric structures 58E20: Harmonic maps [See also 53C43], etc.


Chi, Quo-Shin. The dimension of the moduli space of superminimal surfaces of a fixed degree and conformal structure in the 4-sphere. Tohoku Math. J. (2) 52 (2000), no. 2, 299--308. doi:10.2748/tmj/1178224613.

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