## Smooth compactness of $f$-minimal hypersurfaces with bounded $f$-index

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- by Ezequiel Barbosa, Ben Sharp and Yong Wei PDF
- Proc. Amer. Math. Soc.
**145**(2017), 4945-4961 Request permission

## Abstract:

Let $(M^{n+1},g,e^{-f}d\mu )$ be a complete smooth metric measure space with $2\leq n\leq 6$ and Bakry-Émery Ricci curvature bounded below by a positive constant. We prove a smooth compactness theorem for the space of complete embedded $f$-minimal hypersurfaces in $M$ with uniform upper bounds on $f$-index and weighted volume. As a corollary, we obtain a smooth compactness theorem for the space of embedded self-shrinkers in $\mathbb {R}^{n+1}$ with $2\leq n\leq 6$. We also prove some estimates on the $f$-index of $f$-minimal hypersurfaces.## References

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## Additional Information

**Ezequiel Barbosa**- Affiliation: Departamento de Matemática, Universidade Federal de Minas Gerais (UFMG), Caixa Postal 702, 30123-970, Belo Horizonte, MG, Brazil
- Email: ezequiel@mat.ufmg.br
**Ben Sharp**- Affiliation: Department of Mathematics, University of Warwick, Coventry, CV4 7AL, United Kingdom
- MR Author ID: 1008414
- Email: b.sharp@warwick.ac.uk
**Yong Wei**- Affiliation: Mathematical Sciences Institute, Australian National University, John Dedman Building 27, Union Lane, Canberra ACT 2601, Australia
- MR Author ID: 1036099
- ORCID: 0000-0002-9460-9217
- Email: yong.wei@anu.edu.au
- Received by editor(s): July 11, 2016
- Received by editor(s) in revised form: November 24, 2016
- Published electronically: May 26, 2017
- Additional Notes: The first author was supported by FAPEMIG and CNPq grants. The second author was supported by André Neves’ ERC and Leverhulme trust grants. The third author was supported by Jason D Lotay’s EPSRC grant
- Communicated by: Guofang Wei
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**145**(2017), 4945-4961 - MSC (2010): Primary 53C42, 53C21
- DOI: https://doi.org/10.1090/proc/13628
- MathSciNet review: 3692008