On the moduli space of superminimal surfaces in spheres
type: Item
instance of: scholarly article
title: On the moduli space of superminimal surfaces in spheres
zbMATH Open document ID: 1130.53009
DOI: 10.1155/S0161171203112161
author: Luis Fernández
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2003-12-03T00:00:00Z
full work available at URL: https://eudml.org/doc/53752
review text: It is known [\textit{B. Loo}, Trans. Am. Math. Soc. 313, No. 1, 81--102 (1989; Zbl 0686.58009)] that the dimension of the moduli space of harmonic maps \(f : S^2 \to S^4\) of degree \(d\) in \(S^4\) is \(2d + 4\). In this case the twistor space \({\mathcal L}_2 = \mathrm{SO}(5)/\mathrm{SO}(2)\) is isomorphic to the complex projective space \({\mathbb P}^3\) and so one can characterize twistor lifts of minimal maps as solutions in \({\mathbb P}^3\) of a single differential equation. The paper under review deals with the moduli space of superminimal surfaces into higher dimensional sphere \(S^{2n}\). A superminimal surface in \(S^{2n}\) can be defined as a harmonic map \(f: M_g \to S^{2n}\), where \(M_g\) is a Riemann surface of genus \(g\), whose osculating plane is totally isotropic (i.e., the symmetric complex product of any two vectors in that subspace is zero). In this case the twistor space \({\mathcal L}_n\) of \(S^{2n}\) becomes \(\mathrm{SO}(2n+1)/\mathrm{SO}(n)\) whose dimension is \(\displaystyle{N = \frac{n(n+1)}{2}}\). Although \({\mathcal L}_n\) and \({\mathbb P}^N\) are not isomorphic to each other, they are birationally equivalent. Using a birational correspondence between the twistor space of \(S^{2n}\) and projective space, the author described, up to birational equivalence, the moduli space of superminimal surfaces in \(S^{2n}\) of degree \(d\) as curves of degree \(d\) in projective space satisfying a certain differential system. Using this description, the author also showed that the moduli space of linearly full maps (i.e., not contained in a proper subsphere of \(S^{2n}\)) is nonempty for sufficiently large degree and the dimension of this moduli space for \(n = 3\) and genus \(0\) is greater than or equal to \(2d + 9\). The author also gave a direct, simple proof of the connectedness of the moduli space of superminimal surfaces in \(S^{2n}\) of degree \(d\).
reviewed by: Gabjin Yun
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 2012578
zbMATH Keywords: harmonic surface
zbMATH Keywords: moduli space
zbMATH Keywords: superminimal
zbMATH Keywords: twistor lift
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2012578