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

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Abstract

For a sequence of approximate Dirac-harmonic maps from a closed spin Riemann surface into a stationary Lorentzian manifold with uniformly bounded energy, we study the blow-up analysis and show that the Lorentzian energy identity holds. Moreover, when the targets are static Lorentzian manifolds, we prove the positive energy identity and the no neck property.

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Acknowledgements

The first author was supported by the Fundamental Research Funds for the Central Universities (Grant No. SWU119064). The second author was supported by Shanghai Frontier Research Institute for Modern Analysis (IMA-Shanghai) and Innovation Program of Shanghai Municipal Education Commission (Grant No. 2021-01-07-00-02-E00087). Part of this work was carried out when the first author was a postdoc at the School of Mathematical Sciences, Shanghai Jiao Tong University and later when he was visiting the School of Mathematical Sciences, Shanghai Jiao Tong University. The first author thanks the institution for hospitality and financial support.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

    Wanjun Ai

  2. School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China

    Miaomiao Zhu

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  1. Wanjun Ai
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  2. Miaomiao Zhu
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Correspondence to Miaomiao Zhu.

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Ai, W., Zhu, M. The qualitative behavior for approximate Dirac-harmonic maps into stationary Lorentzian manifolds. Sci. China Math. 65, 1679–1706 (2022). https://doi.org/10.1007/s11425-020-1895-7

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