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

Jiayu Li; Xiangrong Zhu

doi:10.1016/j.anihpc.2018.04.002

JournalsaihpcVol. 36, No. 1pp. 103–118

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

Jiayu Li
College of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, PR China; AMSS, CAS, Beijing 100190, PR China
Xiangrong Zhu
Department of Mathematica, Zhejiang Normal University, Jinhua 321004, PR China

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Abstract

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

α > 1 sup 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]).

Cite this article

Jiayu Li, Xiangrong Zhu, Energy identity and necklessness for a sequence of Sacks–Uhlenbeck maps to a sphere. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 1, pp. 103–118

DOI 10.1016/J.ANIHPC.2018.04.002