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Bubbling analysis for approximate Lorentzian harmonic maps from Riemann surfaces
Xiaoli Han, Jürgen Jost, Lei Liu and Liang Zhao
For a sequence of approximate harmonic maps $(u_n,v_n)$ (meaning that they satisfy the harmonic system up to controlled error terms) from a compact Riemann surface with smooth boundary to a standard static Lorentzian manifold with bounded energy, we prove that identities for the Lorentzian energy hold during the blow-up process. In particular, in the special case where the Lorentzian target metric is of the form $g_N -\beta dt^2$ for some Riemannian metric $g_N$ and some positive function $\beta$ on $N$, we prove that such identities also hold for the positive energy (obtained by changing the sign of the negative part of the Lorentzian energy) and there is no neck between the limit map and the bubbles. As an application, we complete the blow-up picture of singularities for a harmonic map flow into a standard static Lorentzian manifold. We prove that the energy identities of the flow hold at both finite and infinite singular times. Moreover, the no neck property of the flow at infinite singular time is true.