Energy identity and necklessness for a sequence of Sacks-Uhlenbeck maps to a sphere
Let \(M\) be a compact Riemann surface and let \(N\) be a Riemannian manifold. For \(\alpha>1\), the \(\alpha\)-perturbed energy functional for an \(H^{1, 2\alpha}\) function \(u : M \to N\) is defined by \[ E_\alpha(u) : = \frac{1}{2}\int_M \left[(1+|\nabla u|^2 )^\alpha - 1\right]\, dv. \] Note that when \(\alpha=1\), \(E_1 = E\) is the usual energy functional. For any \(\alpha>1\), it is known ([\textit{J. Sacks} and \textit{K. Uhlenbeck}, Ann. Math. (2) 113, 1--24 (1981; Zbl 0462.58014)]) that the \(\alpha\)-perturbed energy functional \(E_\alpha\) satisfies the Palais-Smale condition, and so for each nontrivial homotopy class \([\beta] \in \pi_1(C^0 \cap H^{1, 2}(M, N))\), there is a map \(\gamma^\alpha: [0, 1] \to C^0\cap H^{1, 2\alpha}(M, N)\) with \(\gamma^\alpha \in [\beta]\) such that \[ E_\alpha(\gamma^\alpha(s_\alpha)) = \inf_{\gamma\in [\beta]}\sup_{s\in [0, 1]} E_\alpha(\gamma(s)), \] where \(s_\alpha \in [0, 1]\). Sacks-Uhlenbeck proved that there is a sequence \(\alpha_n \to 1\) as \(n \to \infty\) such that \(u_n:= \gamma^{\alpha_n}(s_{\alpha_n})\) converges to a harmonic map \(u\) outside a finite set of points \(\Gamma\) which is usually called blow-up set. Recalling the maps near every point in \(\Gamma\), one can obtain some harmonic spheres \(\phi_1, \cdots, \phi_m\), and show the following eneryg identity (cf. [\textit{T. H. Colding} and \textit{W. P. Minicozzi II}, Geom. Topol. 12, No. 5, 2537--2586 (2008; Zbl 1161.53352); J. Am. Math. Soc. 18, No. 3, 561--569 (2005; Zbl 1083.53058)], [\textit{T. Lamm}, Trans. Am. Math. Soc. 362, No. 8, 4077--4097 (2010; Zbl 1200.58016)]): \[ \lim_{n\to \infty} E(u_n) = E(u) + \sum_{i=1}^, E(\phi_i). \] In this paper under review, the authors consider a sequence of Sacks-Uhlenbeck maps \(u_\alpha\) which are critical points of \(E_\alpha\) with \[ \sup_{\alpha>1}E_\alpha(u_\alpha) < \infty. \] It is known ([\textit{Y. Li} and \textit{Y. Wang}, ``An example of \(\alpha\)-harmonic map sequence for which energy identity is not true'', \url{arXiv:1303.2862}]) that there is a manifold \(N\) and a sequence of Sacks-Uhlenbeck maps such that the energy identity fails to hold. In this paper, the authors assume that \(N = {\mathbb S}^{k-1}\) is a sphere, and prove the energy identity for a sequence of Sacks-Uhlenbeck maps during blowing up. Furthermore, the authors show, in this case, there is no oscillation on the neck domain. Since the blow-up set is finite, a finite covering \(\{U_i\}\) of \(M\) can be chosen so that the blow-up points are away from the boundaries of \(U_i\). So, it suffices to show the energy identity and necklessness in each \(U_i\). Without loss of generality, \(U_i\) is assumed to be an unit disk \(B_1\) in \({\mathbb R}^2\) with the Euclidean metric.
scientific article; zbMATH DE number 6998053
scientific article