Bubble tree convergence for the harmonic sequence of harmonic surfaces in \(\mathbb{C}\mathbb{P}^n\)
The authors prove a bubble tree convergence results for a map from a compact Riemannian surface into a complex projective space. Let \(f : M \to {\mathbb {CP}^n}\) from a compact Riemannian surface \(M\) with harmonic sequence \(\{f_k\}\) obtained by \(\partial_k f = \partial^k f\) or \(\bar \partial_k f = \bar\partial^k f\). Assume \(f\) is non-isotropic, i.e., the harmonic sequence \(\{f_k\}\) is not finite, or equivalently \(\partial_k f\) (and \(\bar \partial_k f)\) are non-zero holomorphic (anti-holomorphic) map for every \(k \in {\mathbb N}\). With these assumptions, the authors show that there exist a finite set of bubble points \(\{p_1, p_2, \dots, p_l\}\subset M\), a harmonic map \(\hat f : M \to {\mathbb {CP}^n}\) and a finite set of nontrivial harmonic maps (called bubbles) \(h_{\lambda}^\mu : S^2 \to {\mathbb {CP}^n}, \lambda = 1, 2, \dots, l; \mu = 1, 2, \dots, L_\lambda\) such that after selection of a subsequence, \(f_k\) converges uniformly in \(C^\infty\) to \(\hat f\) on \(M \setminus \{p_1, p_2, \dots, p_l\}\) and the following energy identity holds: \[ \lim_{k\to \infty} E(f_k) = E(\hat f) + \sum_{\lambda =1}^l \sum_{\mu = 1}^{L_\lambda} E(h_\lambda^\mu). \]
scientific article; zbMATH DE number 5790101