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Stability and compactness for complete $ f$-minimal surfaces


Authors: Xu Cheng, Tito Mejia and Detang Zhou
Journal: Trans. Amer. Math. Soc. 367 (2015), 4041-4059
MSC (2010): Primary 58J50; Secondary 58E30
DOI: https://doi.org/10.1090/S0002-9947-2015-06207-2
Published electronically: February 18, 2015
MathSciNet review: 3324919
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Abstract: Let $ (M,\bar {g}, e^{-f}d\mu )$ be a complete metric measure space with Bakry-Émery Ricci curvature bounded below by a positive constant. We prove that in $ M$ there is no complete two-sided $ L_f$-stable immersed $ f$-minimal hypersurface with finite weighted volume. Further, if $ M$ is a $ 3$-manifold, we prove a smooth compactness theorem for the space of complete embedded $ f$-minimal surfaces in $ M$ with the uniform upper bounds of genus and weighted volume, which generalizes the compactness theorem for complete self-shrinkers in $ \mathbb{R}^3$ by Colding-Minicozzi.


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Additional Information

Xu Cheng
Affiliation: Instituto de Matematica e Estatística, Universidade Federal Fluminense, Niterói, RJ 24020, Brazil
Email: xcheng@impa.br

Tito Mejia
Affiliation: Instituto de Matematica e Estatística, Universidade Federal Fluminense, Niterói, RJ 24020, Brazil
Email: tmejia.uff@gmail.com

Detang Zhou
Affiliation: Instituto de Matematica e Estatística, Universidade Federal Fluminense, Niterói, RJ 24020, Brazil
Email: zhou@impa.br

DOI: https://doi.org/10.1090/S0002-9947-2015-06207-2
Received by editor(s): March 6, 2013
Published electronically: February 18, 2015
Additional Notes: The first and third authors were partially supported by CNPq and Faperj of Brazil
The second author was supported by CNPq of Brazil
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.