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