Abstract
We derive the Simons-type equation for \(f\)-minimal hypersurfaces in weighted Riemannian manifolds and apply it to obtain a pinching theorem for closed \(f\)-minimal hypersurfaces immersed in the product manifold \(\mathbb {S}^n(\sqrt{2(n-1)})\times \mathbb {R}\) with \(f=\frac{t^2}{4}\). Also, we classify closed \(f\)-minimal hypersurfaces with \(L_f\)-index one immersed in \(\mathbb {S}^n(\sqrt{2(n-1)})\times \mathbb {R}\) with the same \(f\) as above.
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References
Bayle, V.: Propriétés de concavité du profil isopérimétrique et applications, These de Doctorat, (2003)
Cao, H.-D., Li, H.: A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension. Calc. Var. Partial Differ. Eq. 46(3–4), 879–889, (2013) MR 3018176. doi:10.1007/s00526-012-0508-1
Cheng, X., Mejia, T., Zhou, D.: Eigenvalue estimate and compactness for \(f\)-minimal surfaces. Pacific J. Math. (to appear)
Cheng, X., Mejia, T., Zhou, D.: Stability and compactness for complete \(f\)-minimal surfaces. Trans. Amer. Math. Soc. (to appear)
Cheng, X., Zhou, D.: Volume estimate about shrinkers. Proc. Amer. Math. Soc., 141(2), 687–696, (2013). MR 2996973, doi:10.1090/S0002-9939-2012-11922-7
Cheng, X., Zhou, D.: Stability properties and gap theorem for complete \(f\)-minimal hypersurfaces. Bull. Braz. Math. Soc. (to appear)
Colding, T.H., Minicozzi II, W.P.: Generic mean curvature flow I: generic singularities. Ann. Math. 175(2), 755–833 (2012). doi:10.4007/annals.2012.175.2.7, MR 2993752
Espinar, J.M.: Manifolds with density, applications and gradient Schrödinger operators. arXiv:1209.6162v6 (2012)
Fischer-Colbrie, D.: On complete minimal surfaces with finite Morse index in three-manifolds. Invent. Math. 82, 121–132 (1985)
Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Diff. Geom. 31, 285–299 (1990)
Hussey, C.: Classification and analysis of low index mean curvature flow self-shrinkers. arXiv:1303.0354v1
Le, N.Q., Sesum, N.: Blow-up rate of the mean curvature during the mean curvature flow and a gap theorem for self-shrinkers. Comm. Anal. Geom. 19(4), 633–659 (2011). MR 2880211
Li, H., Wei, Y.: \(f\)-minimal surface and manifold with positive \(m\)-Bakry–Émery Ricci curvature. J. Geom. Anal. (2012). arXiv:1209.0895v1 [math.DG]
Liu, G.: Stable weighted minimal surfaces in manifolds with non-negative Bakry-Émery Ricci tensor. Comm. Anal. Geom. 21(5), 1061–1079 (2013)
Lott, J.: Mean curvature flow in a Ricci flow background. Comm. Math. Phys. 313(2), 517–533 (2012). doi:10.1007/s00220-012-1503-2. MR 2942959
Magni, A., Mantegazza, C., Tsatis, E.: Flow by mean curvature inside a moving ambient space. J. Evol. Eq. 13, 561–576 (2013)
Schoen, R., Simon, L., Yau, S.-T.: Curvature estimates for minimal hypersurfaces. Acta Math. 134(3–4), 275–288 (1975). MR 0423263 (54 #11243)
Simons, J.: Minimal varieties in Riemannian manifolds. Ann. Math. 88(2), 62–105 (1968). MR 0233295 (38 #1617)
Acknowledgments
The Xu Cheng and Detang Zhou are partially supported by CNPq and Faperj of Brazil. The Tito Mejia is supported by CNPq of Brazil.
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Communicated by Jiaping Wang.
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Cheng, X., Mejia, T. & Zhou, D. Simons-Type Equation for \(f\)-Minimal Hypersurfaces and Applications. J Geom Anal 25, 2667–2686 (2015). https://doi.org/10.1007/s12220-014-9530-1
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DOI: https://doi.org/10.1007/s12220-014-9530-1