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Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on spheres

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Abstract

We study existence and non-existence of constant scalar curvature metrics conformal and arbitrarily close to homogeneous metrics on spheres, using variational techniques. This describes all critical points of the Hilbert–Einstein functional on such conformal classes, near homogeneous metrics. Both bifurcation and local rigidity type phenomena are obtained for 1-parameter families of U(n + 1), Sp(n + 1) and Spin(9)-homogeneous metrics.

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Authors and Affiliations

  1. Department of Mathematics, University of Notre Dame, 255 Hurley Building, Notre Dame, IN, 46556-4618, USA

    Renato G. Bettiol

  2. Departamento de Matemática, Universidade de São Paulo, Rua do Matão, 1010, São Paulo, SP, 05508-090, Brazil

    Paolo Piccione

Authors
  1. Renato G. Bettiol
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  2. Paolo Piccione
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Correspondence to Renato G. Bettiol.

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Communicated by P. Rabinowitz.

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Bettiol, R.G., Piccione, P. Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on spheres. Calc. Var. 47, 789–807 (2013). https://doi.org/10.1007/s00526-012-0535-y

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  • DOI: https://doi.org/10.1007/s00526-012-0535-y

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