Bubbling analysis for approximate Lorentzian harmonic maps from Riemann surfaces

Xiaoli Han, Jürgen Jost, Lei Liu, and Liang Zhao

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Submission date: 04. May. 2017
Pages: 36
published in: Calculus of variations and partial differential equations, 56 (2017) 6, art-no. 175 
DOI number (of the published article): 10.1007/s00526-017-1271-0
Bibtex
Keywords and phrases: harmonic map, Lorentzian manifold, warped product, blow up, energy identity
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Abstract:
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 − βdt² for some Riemannian metric g_N and some positive function β 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.