STABLE MINIMAL HYPERSURFACES WITH WEIGHTED POINCARÉ INEQUALITY IN A RIEMANNIAN MANIFOLD

Title & Authors
STABLE MINIMAL HYPERSURFACES WITH WEIGHTED POINCARÉ INEQUALITY IN A RIEMANNIAN MANIFOLD
Nguyen, Dinh Sang; Nguyen, Thi Thanh;

Abstract
In this note, we investigate stable minimal hypersurfaces with weighted Poincar$\small{\acute{e}}$ inequality. We show that we still get the vanishing property without assuming that the hypersurfaces is non-totally geodesic. This generalizes a result in [2].
Keywords
minimal hypersurface;stability;weighted Poincar$\small{\acute{e}}$ inequality;
Language
English
Cited by
1.
L2HARMONIC 1-FORMS ON SUBMANIFOLDS WITH WEIGHTED POINCARÉ INEQUALITY, Journal of the Korean Mathematical Society, 2016, 53, 3, 583
References
1.
X. Cheng and D. T. Zhou, Manifolds with weighted Poincare inequality and uniqueness of minimal hypersurfaces, Comm. Anal. Geom. 17 (2009), no. 1, 139-154.

2.
N. T. Dung and K. Seo, Stable minimal hypersurfaces in a Riemannian manifold with pinched negative sectional curvature, Ann. Global Anal. Geom. 41 (2012), no. 4, 447-460.

3.
K. H. Lam, Results on a weighted Poincare inequality of complete manifolds, Trans. Amer. Math. Soc. 362 (2010), no. 10, 5043-5062.

4.
P. F. Leung, An estimate on the Ricci curvature of a submanifold and some applications, Proc. Amer. Math. Soc. 114 (1992), no. 4, 1051-1061.

5.
P. Li and J. Wang, Complete manifolds with positive spectrum, J. Differential Geom. 58 (2001), no. 3, 501-534.

6.
X. Wang, On conformally compact Einstein manifolds, Math. Res. Lett. 8 (2001), no. 5-6, 671-688.