Preprint 26/2016

The qualitative behavior at the free boundary for approximate harmonic maps from surfaces

Jürgen Jost, Lei Liu, and Miaomiao Zhu

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Submission date: 21. Mar. 2016 (revised version: September 2018)
Pages: 39
published in: Mathematische Annalen, 374 (2019) 1-2, p. 133-177 
DOI number (of the published article): 10.1007/s00208-018-1759-8
Keywords and phrases: harmonic map, heat flow, free boundary, blow up, energy identity, no neck
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Let {un} be a sequence of maps from a compact Riemann surface Mwith smooth boundary to a general compact Riemannian manifold N withfree boundary on a smooth submanifold K N satisfying

sup(∥∇un ∥L2(M) +∥τ(un)∥L2(M))≤ Λ,  n

where τ(un) is the tension field of the map un. We show that the energyidentity and the no neck property hold during a blow-up process. Theassumptions are such that this result also applies to the harmonic mapheat flow with free boundary, to prove the energy identity at finite singulartime as well as at infinity time. Also, the no neck property holds at infinitytime.


13.05.2021, 02:16