The dimension of the moduli space of superminimal surfaces of a fixed degree and conformal structure in the 4-sphere
Let \({\mathcal M}_{d,g} (X)\) denote the moduli space of branched superminimal surfaces of a fixed degree \(d\) from any Riemann surface \(X\) of genus \(g\) into the four-sphere. By definition, \({\mathcal M}_{d,g} (X)\) is the variety of all horizontal holomorphic maps from \(X\) into \(\mathbb C \mathbb P^3\) (the twistor space of \(\mathbb S^4\)) satisfying the differential equation \[ z_0dz_1-z_1dz_0+z_2dz_3-z_3dz_2=0, \] where \([z_0:z_1:z_2:z_3]\) are the homogeneus coordinates of \(\mathbb C \mathbb P^3\). In [\textit{Q. S. Chi} and \textit{X. Mo}, Osaka J. Math. 33, 669-696 (1996; Zbl 0902.53043)], it was proved that for any sufficiently large degree non totally geodesic branched superminimal surfaces do exist for any conformal structure and the dimension of each irreducible component of the moduli space is between \(2d-4g+4\) and \(2d-g+4\). In this article, using the canonical contact structure of \(\mathbb C \mathbb P^3\) and thinking of \({\mathcal M}_{d,g} (X)\) as a family of contact maps from \(X\) into \(\mathbb C \mathbb P^3\), the author proves that if \(d=8g+1+3k\), \(k\geq 0\) and \(g\geq 1\), then the dimension of at least one irreducible component of each \({\mathcal M}_{d,g} (X)\) achieves the above lower bound \(2d-g+4\).
scientific article; zbMATH DE number 1533116
scientific article