A space of harmonic maps from the sphere into the complex projective space
type: Item
instance of: scholarly article
zbMATH Open document ID: 1280.58007
DOI: 10.4153/CJM-2012-052-6
author: Hiroko Kawabe
published in: Canadian Journal of Mathematics
publication date: 2013-07-30T00:00:00Z
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 58D27
Mathematics Subject Classification ID: 58D15
Mathematics Subject Classification ID: 53C43
zbMATH DE Number: 6192650
zbMATH Keywords: harmonic maps
zbMATH Keywords: complex projective space
zbMATH Keywords: harmonic sequences
zbMATH Keywords: holomorphic bubble tree
zbMATH Keywords: sphere
MaRDI profile type: Publication
full work available at URL: cjm-2012-052-6
OpenAlex ID: W2907173113
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title: A space of harmonic maps from the sphere into the complex projective space
review text: A map \(f:S^2\to\mathbb{C}P^n\) is harmonic if it is a critical point of the energy functional \(E:W^{1,4}(S^2,\mathbb{C}P^n)\to\mathbb{R}\) defined by \( E(f)=\int_{S^2}|df|^2. \)NEWLINENEWLINEIt is known that the space of harmonic maps from \(S^2\) to \(\mathbb{C}P^n\), denoted by \(\mathrm{Harm}(S^2,\mathbb{C}P^n)\), is the disjoint union of \(\mathrm{Harm}_{k,E}(S^2,\mathbb{C}P^n)\), where the latter denotes the space of harmonic maps of degree \(k\) and energy \(E\). In [\textit{M. A. Guest} and \textit{Y. Ohnita}, J. Math. Soc. Japan 45, No. 4, 671--704 (1993; Zbl 0792.58012)] and [\textit{T. A. Crawford}, Can. Math. Bull. 40, No. 3, 285--295 (1997; Zbl 0889.58028)] it was shown that the space \(\mathrm{Harm}_{k,E}(S^2,\mathbb{C}P^n)\) is path-connected.NEWLINENEWLINEMoreover, let \(\mathrm{Hol}_k(S^2,\mathbb{C}P^n)\) be the space of holomorphic maps of degree \(k>0\) or anti-holomorphic maps if \(k<0\). Let the integer \(r\geq 0\) be the ramification index and consider the space \(\mathrm{Hol}_{k,r}(S^2,\mathbb{C}P^n)\subset\mathrm{Hol}_k(S^2,\mathbb{C}P^2)\).NEWLINENEWLINEFor \(n=2\) the map \( \partial:\mathrm{Hol}_{k,r}(S^2,\mathbb{C}P^n)\to\mathrm{Harm}_{k',E'}(S^2,\mathbb{C}P^n) \) is a homeomorphism [Crawford, loc. cit.] for certain values of \(k'\) and \(E'\). However, for \(n\geq 3\) it is known that the map \(\partial\) is not necessarily continuous.NEWLINENEWLINENevertheless, the main result of the article shows that on certain subspaces of \(\mathrm{Harm}_{k,E}(S^2,\mathbb{C}P^n)\) the map \(\partial\) is continuous also for \(n\geq 3\).NEWLINENEWLINEMore precisely, for \(n\geq 2\) and an \((n-1)\)-tuple \(R_j=(R_0,R_1,\ldots,R_{n-2})\) it is proven that if \(\mathrm{Hol}^*_{k,R_j}(S^2,\mathbb{C}P^n)\) is not empty, then it is a path-connected complex submanifold of \(\mathrm{Hol}^*_{k}(S^2,\mathbb{C}P^n)\). Moreover, it is shown that for \(1\leq s\leq n\) the map \(\partial^s:\mathrm{Hol}^*_{k,R_j}(S^2,\mathbb{C}P^n)\to \mathrm{Harm}_{k,s,R_j}^*(S…
reviewed by: Volker Branding
description: scientific article; zbMATH DE number 6192650