Energy identity and necklessness for a sequence of Sacks–Uhlenbeck maps to a sphere

Li, Jiayu; Zhu, Xiangrong

doi:10.1016/j.anihpc.2018.04.002

Li, Jiayu ; Zhu, Xiangrong

Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 1, pp. 103-118.

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Résumé

Let u be a map from a Riemann surface M to a Riemannian manifold N and $α > 1$ , the α energy functional is defined as

E_{α} (u) = \frac{1}{2} \int_{M} [{(1 + | ▽ u |^{2})}^{α} - 1] d V .

We call $u_{α}$ a sequence of Sacks–Uhlenbeck maps if $u_{α}$ are critical points of $E_{α}$ and

\sup_{α > 1} E_{α} (u_{α}) < \infty .

In this paper, we show the energy identity and necklessness for a sequence of Sacks–Uhlenbeck maps during blowing up, if the target N is a sphere $S^{K - 1}$ . The energy identity can be used to give an alternative proof of Perelman's result [15] that the Ricci flow from a compact orientable prime non-aspherical 3-dimensional manifold becomes extinct in finite time (cf. [3,4]).

MR Zbl

DOI : 10.1016/j.anihpc.2018.04.002

Mots-clés : Energy identity, Sacks–Uhlenbeck map, Harmonic sphere

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     title = {Energy identity and necklessness for a sequence of {Sacks{\textendash}Uhlenbeck} maps to a sphere},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {103--118},
     publisher = {Elsevier},
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Li, Jiayu; Zhu, Xiangrong. Energy identity and necklessness for a sequence of Sacks–Uhlenbeck maps to a sphere. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 1, pp. 103-118. doi : 10.1016/j.anihpc.2018.04.002. http://www.numdam.org/articles/10.1016/j.anihpc.2018.04.002/

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