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Ground state Dirac bubbles and Killing spinors

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Abstract

We prove a classification result for ground state solutions of the critical Dirac equation on \(\mathbb {R}^n\), \(n\geqslant 2\). By exploiting its conformal covariance, the equation can be posed on the round sphere \(\mathbb {S}^n\) and the non-zero solutions at the ground level are given by Killing spinors, up to conformal diffeomorphisms. Moreover, such ground state solutions of the critical Dirac equation are also related to the Yamabe equation for the sphere, for which we crucially exploit some known classification results.

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Notes

  1. This is to identify the spinor bundles associated to different spin structures.

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Acknowledgements

The authors are grateful to B. Ammann for bringing to their attention some results contained in [3] and to G. Buttazzo for pointing out reference [46] to them. A.M. has been partially supported by the project Geometric problems with loss of compactness from Scuola Normale Superiore and by MIUR Bando PRIN 2015 2015KB9WPT\(_{001}\). A.M. and W.B. are members of GNAMPA as part of INdAM and are supported by the GNAMPA 2020 project Aspetti variazionali di alcune PDE in geometria conforme. W.B. and R.W. are supported by Centro di Ricerca Matematica Ennio de Giorgi.

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Authors and Affiliations

  1. Scuola Normale Superiore, Centro De Giorgi, Piazza dei Cavalieri 3, 56100, Pisa, Italy

    William Borrelli

  2. Scuola Normale Superiore, Piazza dei Cavalieri 7, 56100, Pisa, Italy

    Andrea Malchiodi

  3. International School for Advanced Studies (SISSA), Via Bonomea 265, 34136, Trieste, Italy

    Ruijun Wu

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  1. William Borrelli
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  2. Andrea Malchiodi
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Correspondence to William Borrelli.

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Communicated by P. Chrusciel.

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Borrelli, W., Malchiodi, A. & Wu, R. Ground state Dirac bubbles and Killing spinors. Commun. Math. Phys. 383, 1151–1180 (2021). https://doi.org/10.1007/s00220-021-04013-1

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