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Stability and geometric properties of constant weighted mean curvature hypersurfaces in gradient Ricci solitons

Abstract

In this paper, we study stability properties of hypersurfaces with constant weighted mean curvature (CWMC) in gradient Ricci solitons. The CWMC hypersurfaces generalize the f-minimal hypersurfaces and appear naturally in the isoperimetric problems in smooth metric measure spaces. We obtain a result about the relationship between the properness and extrinsic volume growth under the assumption of a limitation for the weighted mean curvature of the immersion. Moreover, we estimate Morse index for CWMC hypersurfaces in terms of the dimension of the space of parallel vector fields restricted to hypersurface.

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Correspondence to Hilário Alencar.

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The Hilário Alencar was partially supported by CNPq of Brazil. The Adina Rocha was partially supported by CAPES of Brazil.

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Alencar, H., Rocha, A. Stability and geometric properties of constant weighted mean curvature hypersurfaces in gradient Ricci solitons. Ann Glob Anal Geom 53, 561–581 (2018). https://doi.org/10.1007/s10455-017-9588-7

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  • DOI: https://doi.org/10.1007/s10455-017-9588-7

Keywords

  • Hypersurface
  • Weighted volume
  • Weighted mean curvature
  • Stability
  • Index

Mathematics Subject Classification

  • 58J50
  • 53C42
  • 58E30