Let $$\{u_n\}$${un} be a sequence of maps from a compact Riemann surface M with smooth boundary to a general compact Riemannian manifold N with free boundary on a smooth submanifold $$K\subset N$$K⊂N satisfying $$\begin{aligned} \sup _n \ \left( \Vert \nabla u_n\Vert _{L^2(M)}+\Vert \tau (u_n)\Vert _{L^2(M)}\right) \le \Lambda , \end{aligned}$$supn‖∇un‖L2(M)+‖τ(un)‖L2(M)≤Λ,where $$\tau (u_n)$$τ(un) is the tension field of the map $$u_n$$un. We show that the energy identity and the no neck property hold during a blow-up process. The assumptions are such that this result also applies to the harmonic map heat flow with free boundary, to prove the energy identity at finite singular time as well as at infinity time. Also, the no neck property holds at infinity time.

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表面近似调和映射的自由边界处的定性行为

令 $$\{u_n\}$${un} 是从具有平滑边界的紧致黎曼曲面 M 到在光滑子流形 $$K\subset N$$K 上具有自由边界的一般紧致黎曼流形 N 的映射序列⊂N 满足 $$\begin{aligned} \sup _n \ \left( \Vert \nabla u_n\Vert _{L^2(M)}+\Vert \tau (u_n)\Vert _{L^2(M )}\right) \le \Lambda , \end{aligned}$$supn‖∇un‖L2(M)+‖τ(un)‖L2(M)≤Λ，其中 $$\tau (u_n)$$ τ(un) 是地图 $$u_n$$un 的张力场。我们表明，在爆炸过程中，能量恒等式和无颈属性保持不变。假设该结果也适用于具有自由边界的谐波映射热流，以证明有限奇异时间和无限时间的能量恒等式。此外，无颈属性在无限时间保持不变。