# 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

## 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

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

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