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Isoperimetric inequality and Weitzenböck type formula for critical metrics of the volume

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Abstract

We provide an isoperimetric inequality for critical metrics of the volume functional with nonnegative scalar curvature on compact manifolds with boundary. In addition, we establish a Weitzenböck type formula for critical metrics of the volume functional on four-dimensional manifolds. As an application, we obtain a classification result for such metrics.

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Acknowledgement

The authors want to thank the referee for his careful reading, relevant remarks and valuable suggestions. Moreover, the authors want to thank R. Batista and P. Wu for helpful conversations about this subject.

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Authors and Affiliations

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

    Halyson Baltazar

  2. Instituto de Ciências Exatas e da Natureza, UNILAB, 62785-000, Acarape, Ceará, Brazil

    Rafael Diógenes

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

    Ernani Ribeiro Jr.

Authors
  1. Halyson Baltazar
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  2. Rafael Diógenes
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  3. Ernani Ribeiro Jr.
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Corresponding author

Correspondence to Ernani Ribeiro Jr..

Additional information

H. Baltazar was partially supported by CNPq/Brazil and FAPEPI/Brazil

E. Ribeiro Jr was partially supported by grants from CNPq/Brazil (Grant: 303091/2015-0), PRONEX-FUNCAP/CNPq/Brazil and CAPES/Brazil - Finance Code 001.

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Baltazar, H., Diógenes, R. & Ribeiro, E. Isoperimetric inequality and Weitzenböck type formula for critical metrics of the volume. Isr. J. Math. 234, 309–329 (2019). https://doi.org/10.1007/s11856-019-1930-2

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  • DOI: https://doi.org/10.1007/s11856-019-1930-2

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