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Bifurcation for the Constant Scalar Curvature Equation and Harmonic Riemannian Submersions

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Abstract

We study bifurcation for the constant scalar curvature equation along a one-parameter family of Riemannian metrics on the total space of a harmonic Riemannian submersion. We provide an existence theorem for bifurcation points and a criterion to see that the conformal factors corresponding to the bifurcated metrics must be indeed constant along the fibers. In the case of the canonical variation of a Riemannian submersion with totally geodesic fibers, we characterize discreteness of the set of all degeneracy points along the family and give a sufficient condition to guarantee that bifurcation necessarily occurs at every point where the linearized equation has a nontrivial solution. In the model case of quaternionic Hopf fibrations, we show that \({{\,\mathrm{SU}\,}}(2)\)-symmetry-breaking bifurcation does not occur except at the round metric.

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Acknowledgements

N. Otoba is supported by the DFG (Deutsche Forschungsgemeinschaft), SFB 1085 Higher Invariants. J. Petean is supported by Grant 220074 Fondo Sectorial de Investigación para la Educación CONACYT. We thank the referees for their helpful comments and suggestions that improved the text.

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Authors and Affiliations

  1. Universität Regensburg, 93040, Regensburg, Deutschland

    Nobuhiko Otoba

  2. Centro de Investigación en Matemáticas, Jalisco S/N, Col. Valenciana, CP: 36023, Guanajuato, Gto, Mexico

    Jimmy Petean

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  1. Nobuhiko Otoba
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  2. Jimmy Petean
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Correspondence to Nobuhiko Otoba.

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Otoba, N., Petean, J. Bifurcation for the Constant Scalar Curvature Equation and Harmonic Riemannian Submersions. J Geom Anal 30, 4453–4463 (2020). https://doi.org/10.1007/s12220-019-00265-5

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  • DOI: https://doi.org/10.1007/s12220-019-00265-5

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