Stability and compactness for complete $f$-minimal surfaces
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- by Xu Cheng, Tito Mejia and Detang Zhou PDF
- Trans. Amer. Math. Soc. 367 (2015), 4041-4059 Request permission
Abstract:
Let $(M,\overline {g}, e^{-f}d\mu )$ 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.References
- Huai-Dong Cao and Detang Zhou, On complete gradient shrinking Ricci solitons, J. Differential Geom. 85 (2010), no. 2, 175–185. MR 2732975
- Xu Cheng, Tito Mejia, and Detang Zhou, Eigenvalue estimate and compactness for closed $f$-minimal surfaces, Pacific J. Math. 271 (2014), no. 2, 347–367. MR 3267533, DOI 10.2140/pjm.2014.271.347
- Xu Cheng and Detang Zhou, Volume estimate about shrinkers, Proc. Amer. Math. Soc. 141 (2013), no. 2, 687–696. MR 2996973, DOI 10.1090/S0002-9939-2012-11922-7
- Tobias H. Colding and William P. Minicozzi II, Smooth compactness of self-shrinkers, Comment. Math. Helv. 87 (2012), no. 2, 463–475. MR 2914856, DOI 10.4171/CMH/260
- Tobias H. Colding and William P. Minicozzi II, Generic mean curvature flow I: generic singularities, Ann. of Math. (2) 175 (2012), no. 2, 755–833. MR 2993752, DOI 10.4007/annals.2012.175.2.7
- Tobias Holck Colding and William P. Minicozzi II, A course in minimal surfaces, Graduate Studies in Mathematics, vol. 121, American Mathematical Society, Providence, RI, 2011. MR 2780140, DOI 10.1090/gsm/121
- Tobias H. Colding and William P. Minicozzi II, Estimates for parametric elliptic integrands, Int. Math. Res. Not. 6 (2002), 291–297. MR 1877004, DOI 10.1155/S1073792802106106
- Tobias H. Colding and William P. Minicozzi II, Embedded minimal surfaces without area bounds in 3-manifolds, Geometry and topology: Aarhus (1998), Contemp. Math., vol. 258, Amer. Math. Soc., Providence, RI, 2000, pp. 107–120. MR 1778099, DOI 10.1090/conm/258/04058
- Qi Ding and Y. L. Xin, Volume growth, eigenvalue and compactness for self-shrinkers, Asian J. Math. 17 (2013), no. 3, 443–456. MR 3119795, DOI 10.4310/AJM.2013.v17.n3.a3
- Frank Morgan, Manifolds with density, Notices Amer. Math. Soc. 52 (2005), no. 8, 853–858. MR 2161354
- Ovidiu Munteanu and Jiaping Wang, Analysis of weighted Laplacian and applications to Ricci solitons, Comm. Anal. Geom. 20 (2012), no. 1, 55–94. MR 2903101, DOI 10.4310/CAG.2012.v20.n1.a3
- Ovidiu Munteanu and Jiaping Wang, Geometry of manifolds with densities, Adv. Math. 259 (2014), 269–305. MR 3197658, DOI 10.1016/j.aim.2014.03.023
- Guofang Wei and Will Wylie, Comparison geometry for the Bakry-Emery Ricci tensor, J. Differential Geom. 83 (2009), no. 2, 377–405. MR 2577473, DOI 10.4310/jdg/1261495336
- B. White, Curvature estimates and compactness theorems in $3$-manifolds for surfaces that are stationary for parametric elliptic functionals, Invent. Math. 88 (1987), no. 2, 243–256. MR 880951, DOI 10.1007/BF01388908
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
- 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 - © Copyright 2015
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3324919