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A perturbation method for spinorial Yamabe type equations on \(S^m\) and its application

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Abstract

For \(m\ge 2\), we prove the existence of non-trivial solutions for a certain kind of nonlinear Dirac equations with critical Sobolev nonlinearities on \(S^m\) via a perturbative variational method. For the special case \(m=2\), this establishes the existence of a conformal immersion \(S^2\rightarrow \mathbb R ^3\) with prescribed mean curvature \(H\) which is close to a positive constant under an index counting condition on \(H\).

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Acknowledgments

I would like to express my gratitude to the anonymous referee for drawing my attention to the four vertex theorem and the recent result of M. Anderson [11]. He also gave me a lot of helpful suggestions that made this paper much easier to read.

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  1. Department of Mathematics, Graduate School of Science and Engineering, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo, 152-8551, Japan

    Takeshi Isobe

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  1. Takeshi Isobe
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Isobe, T. A perturbation method for spinorial Yamabe type equations on \(S^m\) and its application. Math. Ann. 355, 1255–1299 (2013). https://doi.org/10.1007/s00208-012-0818-9

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  • DOI: https://doi.org/10.1007/s00208-012-0818-9

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