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

Assume that M is a closed surface and N is a compact Riemannian manifold without boundary. Let uα : M → N be the critical point of Eα with Eα(uα) < C. Assume u0 is the weak limit of uα in W (M,N) and x1 is the only blow-up point in Bσ(x1) ⊂ M with n0 bubbles. Then, on a local coordinate system on Bσ(x1) which origin is x1, we can find sequences xiα → 0, λiα → 0 (i = 1, · · · , n0) s.t. uα(xiα + λiαx) → v, where v are harmonic maps from S to N . We define μi = lim inf α→1 (λiα) . We will prove that lim α→1 Eα(uα, Bσ(x1)) = E(u0, Bσ(x1)) + |Bσ(x1)|+ n0 ∑ j=1 μ2jE(v ). Further, when n0 = 1, we define ν = lim inf α→1 (λ1α) − √ , then we have: If ν = 1, then u0(Bσ(x1)) ∪ v(S) is connected; If 1 < ν < +∞, then u0(Bσ(x1)) and v(S) are connected by a geodesic with length L = √ E(v) π log ν. If ν = +∞, the neck contains at least one geodesic with infinite length. We also give an example of neck which shows the neck contains at least one geodesic of infinite length. Mathematics Subject Classification: 58E20, 35J60.