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On Global Dynamics of Schrödinger Map Flows on Hyperbolic Planes Near Harmonic Maps

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Abstract

The results of this paper are twofold. In the first part, we prove that for Schrödinger map flows from hyperbolic planes to Riemannian surfaces with non-positive sectional curvatures, the harmonic maps which are holomorphic or anti-holomorphic of arbitrary size are asymptotically stable. In the second part, we prove that for Schrödinger map flows from hyperbolic planes into Kähler manifolds, the admissible harmonic maps of small size are asymptotically stable. The asymptotic stability results stated here contain two types: one is the convergence in \(L^{\infty }_x\) as the previous works, the other is convergence to harmonic maps plus radiation terms in the energy space, which is new in literature of Schrödinger map flows without symmetric assumptions.

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Acknowledgements

The author owes sincere gratitude to the referees for the insightful comments which intensely improved the presentation of this work. This work was partially supported by NSF-China Grant-1200010237 and the Natural Science Foundation of Zhejiang Province under Grant No. LY22A010005.

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  1. School of Mathematics and Statistics, Ningbo University, Ningbo, 315211, Zhejiang, People’s Republic of China

    Ze Li

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Correspondence to Ze Li.

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Communicated by A. Ionescu.

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Li, Z. On Global Dynamics of Schrödinger Map Flows on Hyperbolic Planes Near Harmonic Maps. Commun. Math. Phys. 393, 279–345 (2022). https://doi.org/10.1007/s00220-022-04368-z

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