Energy identity and necklessness for a sequence of Sacks-Uhlenbeck maps to a sphere
type: Item
instance of: scholarly article
title: Energy identity and necklessness for a sequence of Sacks-Uhlenbeck maps to a sphere
zbMATH Open document ID: 1410.58004
author: Jiayu Li
author: Xiangrong Zhu
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2019-01-08T00:00:00Z
review text: 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 …
reviewed by: Gabjin Yun
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 53C43
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 6998053
zbMATH Keywords: harmonic map
zbMATH Keywords: energy identity
zbMATH Keywords: harmonic sphere
zbMATH Keywords: perturbed energy functional
zbMATH Keywords: Sacks-Uhlenbeck map
MaRDI profile type: Publication
full work available at URL: j.anihpc.2018.04.002
OpenAlex ID: W2797651686
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DOI: 10.1016/J.ANIHPC.2018.04.002
Wikidata QID: Q129986822
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description: scientific article; zbMATH DE number 6998053