Smooth compactness of -minimal hypersurfaces with bounded -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 be a complete smooth metric measure space with and Bakry-Émery Ricci curvature bounded below by a positive constant. We prove a smooth compactness theorem for the space of complete embedded -minimal hypersurfaces in with uniform upper bounds on -index and weighted volume. As a corollary, we obtain a smooth compactness theorem for the space of embedded self-shrinkers in with . We also prove some estimates on the -index of -minimal hypersurfaces.

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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

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