Volume Functional of Compact 4-Manifolds with a Prescribed Boundary Metric

Abstract

We prove that a critical metric of the volume functional on a 4-dimensional compact manifold with boundary satisfying a second-order vanishing condition on the Weyl tensor must be isometric to a geodesic ball in a simply connected space form \(\mathbb {R}^{4}\), \(\mathbb {H}^{4}\) or \(\mathbb {S}^{4}.\) Moreover, we provide an integral curvature estimate involving the Yamabe constant for critical metrics of the volume functional, which allows us to get a rigidity result for such critical metrics.

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Acknowledgements

The authors want to thank the referee for careful reading, relevant remarks and valuable suggestions. Moreover, they want to thank A. Barros, E. Barbosa and R. Batista for fruitful conversations about this subject.

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Correspondence to E. Ribeiro Jr..

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

E. Ribeiro was partially supported by CNPq/Brazil [Grant: 305410/2018-0 and 160002/2019-2], PRONEX - FUNCAP /CNPq/ Brazil and CAPES/ Brazil - Finance Code 001.

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Baltazar, H., Diógenes, R. & Ribeiro, E. Volume Functional of Compact 4-Manifolds with a Prescribed Boundary Metric. J Geom Anal (2020). https://doi.org/10.1007/s12220-020-00452-9

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Keywords

  • Volume functional
  • Critical metrics
  • Compact manifolds
  • Boundary

Mathematics Subject Classification

  • Primary 53C25
  • 53C20
  • 53C21
  • Secondary 53C65