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


Authors: Ezequiel Barbosa, Ben Sharp and Yong Wei
Journal: Proc. Amer. Math. Soc. 145 (2017), 4945-4961
MSC (2010): Primary 53C42, 53C21
DOI: https://doi.org/10.1090/proc/13628
Published electronically: May 26, 2017
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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.


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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
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
Email: yong.wei@anu.edu.au

DOI: https://doi.org/10.1090/proc/13628
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
Article copyright: © Copyright 2017 American Mathematical Society