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On static manifolds and related critical spaces with zero radial Weyl curvature

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Abstract

The aim of this paper is to study compact Riemannian manifolds \((M,\,g)\) that admit a non-constant solution to the system of equations

$$\begin{aligned} -\Delta f\, g+Hess f-fRic=\mu Ric+\lambda g, \end{aligned}$$

where Ric is the Ricci tensor of g whereas \(\mu \) and \(\lambda \) are two real parameters. More precisely, under assumption that \((M,\,g)\) has zero radial Weyl curvature, this means that the interior product of \(\nabla f\) with the Weyl tensor W is zero, we shall provide the complete classification for the following structures: positive static triples, critical metrics of volume functional and critical metrics of the total scalar curvature functional.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal do Piauí- UFPI, Campus Petrônio Portella, Teresina, Piauí, 64049-550, Brazil

    H. Baltazar & R. Batista

  2. Departamento de Matemática, Universidade Federal do Ceará - UFC, Campus do Pici, Av. Humberto Monte, Bloco 914, Fortaleza, CE, 60455-760, Brazil

    A. Barros

  3. Instituto Federal de Educação, Ciência e Tecnologia do Ceará (IFCE), Campus Caucaia, Caucaia, CE, 61609-090, Brazil

    E. Viana

Authors
  1. H. Baltazar
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  2. A. Barros
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  3. R. Batista
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  4. E. Viana
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Correspondence to H. Baltazar.

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Communicated by Andreas Cap.

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The first author was partially supported by PPP/FAPEPI/MCT/CNPq, Brazil, Grant 007/2018. The second author was partially supported by CNPq, Brazil, Grant 307514/2017-0. The third author was partially supported by CNPq, Brazil, Grant 310881/2017-0.

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Baltazar, H., Barros, A., Batista, R. et al. On static manifolds and related critical spaces with zero radial Weyl curvature. Monatsh Math 191, 449–463 (2020). https://doi.org/10.1007/s00605-019-01365-8

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  • DOI: https://doi.org/10.1007/s00605-019-01365-8

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