The dimension of the moduli space of superminimal surfaces of a fixed degree and conformal structure in the 4-sphere
type: Item
instance of: scholarly article
title: The dimension of the moduli space of superminimal surfaces of a fixed degree and conformal structure in the 4-sphere
zbMATH Open document ID: 0997.53043
author: Quo-Shin Chi
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2002-11-10T00:00:00Z
review text: 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 \(2d-g+4\).
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 1533116
zbMATH Keywords: moduli space
zbMATH Keywords: superminimal surfaces in the 4-sphere
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178224613
OpenAlex ID: W2095587176
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DOI: 10.2748/TMJ/1178224613
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description: scientific article; zbMATH DE number 1533116