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A spinorial energy functional: critical points and gradient flow

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Abstract

Let M be a compact spin manifold. On the universal bundle of unit spinors we study a natural energy functional whose critical points, if \(\dim M \ge 3\), are precisely the pairs \((g,{\varphi })\) consisting of a Ricci-flat Riemannian metric g together with a parallel g-spinor \({\varphi }\). We investigate the basic properties of this functional and study its negative gradient flow, the so-called spinor flow. In particular, we prove short-time existence and uniqueness for this flow.

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Acknowledgments

The authors thank the referees for carefully reading the manuscript which led to considerable improvements of the text.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Regensburg, Universitätsstrasse 40, 93040, Regensburg, Germany

    Bernd Ammann

  2. Mathematisches Seminar der Universität Kiel, Ludewig-Meyn Strasse 4, 24098, Kiel, Germany

    Hartmut Weiss

  3. Institut für Geometrie und Topologie der Universität Stuttgart, Pfaffenwaldring 57, 70569, Stuttgart, Germany

    Frederik Witt

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  1. Bernd Ammann
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  2. Hartmut Weiss
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  3. Frederik Witt
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Correspondence to Bernd Ammann.

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Ammann, B., Weiss, H. & Witt, F. A spinorial energy functional: critical points and gradient flow. Math. Ann. 365, 1559–1602 (2016). https://doi.org/10.1007/s00208-015-1315-8

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  • DOI: https://doi.org/10.1007/s00208-015-1315-8

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