Asymptotic analysis for Dirac-harmonic maps from degenerating spin surfaces and with bounded index

Jürgen Jost, Lei Liu, and Miaomiao Zhu

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Submission date: 23. May. 2018 (revised version: May 2019)
Pages: 36
published in: Calculus of variations and partial differential equations, 58 (2019) 4, art-no. 142 
DOI number (of the published article): 10.1007/s00526-019-1557-5
Bibtex
MSC-Numbers: 53C43, 58E20
Keywords and phrases: Dirac-harmonic maps, degenerating spin surface, blow-up, neck analysis, index, Pohozaev type constant
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Abstract:
We study the refined blow-up behaviour of a sequence of Dirac-harmonic maps from degenerating spin surfaces with uniformly bounded energy in the case that the domain surfaces converge to a spin surface with only Neveu-Schwarz type nodes. For Dirac-harmonic necks appearing near the nodes, we show that the limit of the map part of each neck is a geodesic in the target manifold. Moreover, we give a length formula for the limit geodesics appearing near the node in terms of the Pohozaev type constants associated to the sequence. In particular, if the Ricci curvature of the target manifold has a positive lower bound and the Dirac-harmonic sequence has bounded index, then the limit of the map part of the necks consist of geodesics of finite length and the energy identities hold.