Harmonic maps from the Riemann sphere into the complex projective space and the harmonic sequences

Abstract

When harmonic maps from the Riemann sphere into the complex projective space are energy bounded, it contains a subsequence converging to a bubble tree map f^I: T^I → CPⁿ. We show that their ∂-transforms and $\overline{\partial}$-transforms are also energy bounded. Hence their subsequences converge to harmonic bubble tree maps $f_1^{I_1}:T^{I_1}$ → CPⁿ and $f_{-1}^{I_{-1}}:T^{I_{-1}}$ → CPⁿ respectively. In this paper, we show relations between f^I, $f_1^{I_1}$ and $f_{-1}^{I_{-1}}$.