The moduli space of branched superminimal surfaces of a fixed degree, genus and conformal structure in the four-sphere
Let \(N\) be an oriented Riemannian \(4\)-manifold and denote by \(Z_\pm \) the bundle of pointwise complex structures on \(N\). Let \(f:M\rightarrow N\) be an immersion, where \(M\) is a compact Riemann surface with induced metric. The twistor lifts \(\widetilde f_+:M\rightarrow Z_+,\widetilde f_-:M\rightarrow Z_-\) can be obtained from \(f\) by using the techniques of twistor theory. The immersion \(f\) is superminimal with positive spin (with negative spin) if \(f\) is minimal and \(\widetilde f_+\) (\(\widetilde f_-\)) is \(J\)-holomorphic, where \(J\) is the Hermitian structure on \(Z\), defined by the splitting of the tangent bundle to \(Z\) into the vertical and horizontal distributions (the last one being determined by the Levi-Civita connection on \(N\)). In the case where \(N=S^4\), the expressions of \(\widetilde f_\pm \) can be made explicit. The authors study branched superminimal immersions of arbitrary degree from \(M\) to \(S^4\). Theorem 1. Let \(M\) be a Riemann surface of genus \(g\geq 1\). Then all the branched superminimal immersions of degree \(d\leq 5\) from \(M\) to \(S^4\) are totally geodesic. Theorem 2. \(M\) admits a nontotally geodesic branched superminimal immersion of degree \(6\) into \(S^4\) if and only if \(M\) is hyperelliptic. Theorem 3. Assume that \(g\geq 3\). Then \({\mathcal M}_6(M)= V_1\cup V_2\), where \(V_1\) is the totally geodesic part and \(V_2\) is isomorphic to the nontotally geodesic part of \({\mathcal M}_3({\mathbb{C} P}^1)\). \(V_1\) and \(V_2\) are identified along the singular locus of \({\mathcal M}_3({\mathbb{C} P}^1)\). In particular, nontotally geodesic branched superminimal immersions of degree \(6\) from \(M\) into \(S^4\) are the pullback of nontotally geodesic branched superminimal spheres of degree \(3\) via the branched double covering of \(M\) into \({\mathbb{C} P}^1\). Furthermore, \({\mathcal M}_6(M)\simeq {\mathcal M}_3({\mathbb{C} P}^1)\) only when \(g\geq 6\). Theorem 4. Assume that \(g\geq 2\) (\(g=1,\) respectively). If \(d\geq 5g+4\) (\(\geq 6\), respectively), then there is a nontotally geodesic branched superminimal immersion of degree \(d\) from \(M\) to \(S^4\). The immersion is generically one-to-one. Theorem 5. If \(M\) is a generic Riemann surface of genus \(g\), then the dimension of each irreducible component of \({\mathcal M}_d (M)\) is between \(2d-4g+4\) and \(2d-g+4\), where the upper bound is achieved by the totally geodesic component. Some remarks are made in the case where \(g=1\).
scientific article; zbMATH DE number 1018459