A weak energy identity and the length of necks for a sequence of Sacks–Uhlenbeck α-harmonic maps☆

By Yuxiang Li, Younde Wang. 2010

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Abstract In this paper we discuss the convergence behavior of a sequence of α-harmonic maps u α with E α ( u α ) C from a compact surface ( M , g ) into a compact Riemannian manifold ( N , h ) without boundary. Generally, such a sequence converges weakly to a harmonic map in W 1 , 2 ( M , N ) and strongly in C ∞ away from a finite set of points in M. If energy concentration phenomena appears, we show a generalized energy identity and discover a direct convergence relation between the blow-up radius and the parameter α which ensures the energy identity and no-neck property. We show that the necks converge to some geodesics. Moreover, in the case there is only one bubble, a length formula for the neck is given. In addition, we also give an example which shows that the necks contain at least a geodesic of infinite length.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/36394275ba0c070c07607620be56957efd9c6afe