Bubbling analysis for approximate Lorentzian harmonic maps from Riemann surfaces

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 −βdt 2 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.

The qualitative behavior for approximate Dirac-harmonic maps into stationary Lorentzian manifolds

“…In this paper, we shall answer the above two questions for (approximate) Dirac-harmonic maps into certain Lorentzian manifolds, which generalize the results in the case of harmonic maps [12,13].…”

A global weak solution to the Lorentzian harmonic map flow

“…Combing this with the small energy regularity theory for approximate Lorentzian harmonic maps (see Lemma 2.1 and Lemma 2.2 in [10]), we know that there exist a positive constant ǫ ′ depending only on λ, Λ, N × R, a finite points set…”

Lorentzian harmonic map and its heat flow

“…另外, 我们还期望了解爆破出的泡 泡之间是否有脖子连接并估计脖子的长度. 当目标流形是稳态流形时, 这些问题比较复杂, 所以, Han 等 [49] 同时研究了稳态和静态流形两种情形, 他们对目标流形为稳态流形的渐近 Lorentz 调和映射序 列的爆破行为给出了部分刻画, 对静态情形给出了完整刻画. 渐近 Lorentz 调和映射序列的相关结果 更为一般, 可以自然地推广到热流的情形.…”

“…渐近 Lorentz 调和映射序列的相关结果 更为一般, 可以自然地推广到热流的情形. 在文献 [49] 中, 因为还要处理边界爆破的情形, 所以, 我们 舍弃了 Hopf 微分的办法, 取而代之的是通过对 Lorentz 调和映射方程结构的观察, 利用分部积分得到 某种 Pohozaev 型等式, 由此可以给出径向能量与切向能量之间的关系. 在经典调和映射的爆破分析 中, Pohozaev 型等式最早由 Wang 和 Lin [50] 引入.…”

“…定理 3.6 可以看成是在定理 2.3 的基础上的推广, 对张量场非零的情形同样证明了 Lorentz 能量 的能量等式. 当目标流行是静态流形时, 分析会化简很多, 对于爆破行为的认识也更清晰, 文献 [49] 还 证明了如下结论.…”