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Harmonic maps from the Riemann sphere into the complex projective space and the harmonic sequences. (English) Zbl 1214.58006
Due to Sacks and Uhlenbeck, any harmonic map defined on a closed surface with bounded energy contains a subsequence weakly converging to a set of harmonic maps and a bubble phenomenon may occur in the convergence.
In the paper under review, the author concentrates on harmonic maps from the Riemann sphere \((S^2, g_0)\), identified with \({\mathbb {CP}^1}\), into the complex projective space \({\mathbb {CP}^n}\). Combining the results due to J. Eells and J. C. Wood [Adv. Math. 49, 217–263 (1983; Zbl 0528.58007)] with the ones of J. G. Wolfson [J. Differ. Geom. 27, No. 1, 161–178 (1988; Zbl 0642.58021)], the author obtains, for each harmonic map \(f : S^2 \to {\mathbb {CP}^n}\), harmonic sequences under \(\partial\)-transform and \(\bar \partial\)-transform. The author proves that, if a harmonic map from the Riemann sphere into the complex projective space has bounded energy, then their \(\partial\)-transforms and \(\bar \partial\)-transforms also have bounded energy. This implies that their subsequences converge to harmonic bubble tree maps, respectively.
MSC:
58E20 Harmonic maps, etc.
53C43 Differential geometric aspects of harmonic maps
30F15 Harmonic functions on Riemann surfaces
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