A space of harmonic maps from the sphere into the complex projective space (Q2841824)

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 \(\partial^s:\mathrm{Hol}^*_{k,R_j}(S^2,\mathbb{C}P^n)\to \mathrm{Harm}_{k,s,R_j}^*(S^2,\mathbb{C}P^n)\) is a homeomorphism, where \(\mathrm{Harm}_{k,s,R_j}^*(S^2,\mathbb{C}P^n)\) consists of anti-holomorphic maps. In addition, this implies that for \(0\leq s\leq n-1\) the map \( \partial:\mathrm{Harm}^*_{k,s,R_j}(S^2,\mathbb{C}P^n)\to \mathrm{Harm}^*_{k,s+1,R_j} \) is a homeomorphism.NEWLINENEWLINEThe paper gives the necessary background on harmonic maps into complex projective space and their harmonic sequences. In addition, several facts about the bubble tree convergence for harmonic maps are recalled. Based on the methods from [\textit{T. H. Parker}, J. Differ. Geom. 44, No. 3, 595--633 (1996; Zbl 0874.58012)] a gluing theorem of a holomorphic bubble tree map is established. Finally, the last section of the paper gives several examples illustrating the results.