The moduli space of branched superminimal surfaces of a fixed degree, genus and conformal structure in the four-sphere
type: Item
instance of: scholarly article
title: The moduli space of branched superminimal surfaces of a fixed degree, genus and conformal structure in the four-sphere
zbMATH Open document ID: 0902.53043
author: Quo-Shin Chi
author: Xiaokang Mo
published in: Osaka Journal of Mathematics
publication date: 1997-07-10T00:00:00Z
review text: 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 \(g\geq 2\) (\…
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 58D27
zbMATH DE Number: 1018459
zbMATH Keywords: minimal immersions
zbMATH Keywords: moduli space
zbMATH Keywords: twistor space of a four-manifold
zbMATH Keywords: superminimal immersions
zbMATH Keywords: Riemann surfaces
reviewed by: Vasile Oproiu
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1018459