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Estimates for the first eigenvalue of the drifting Laplace and the p-Laplace operators on submanifolds with bounded mean curvature in the hyperbolic space.
- Source :
-
Journal of Mathematical Analysis & Applications . Dec2017, Vol. 456 Issue 2, p787-795. 9p. - Publication Year :
- 2017
-
Abstract
- In this paper, we successfully give two interesting lower bounds for the first eigenvalue of submanifolds (with bounded mean curvature) in a hyperbolic space. More precisely, let M be an n -dimensional complete noncompact submanifold in a hyperbolic space and the norm of its mean curvature vector ‖ H ‖ satisfies ‖ H ‖ ⩽ α < n − 1 , then we prove that the first eigenvalue λ 1 , p ( M ) of the p -Laplacian Δ p on M satisfies λ 1 , p ( M ) ⩾ ( n − 1 − α p ) p , 1 < p < ∞ , with equality achieved when M is totally geodesic and p = 2 ; let ( M , g , e − φ d v g ) be an n -dimensional complete noncompact smooth metric measure space with M being a submanifold in a hyperbolic space, and ‖ H ‖ ⩽ α < n − 1 , ‖ ∇ φ ‖ ⩽ C with ∇ the gradient operator on M , then we show that the first eigenvalue λ 1 , φ ( M ) of the weighted Laplacian Δ φ on M satisfies λ 1 , φ ( M ) ⩾ ( n − 1 − α − C ) 2 4 , with equality attained when M is totally geodesic and φ = constant . [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 0022247X
- Volume :
- 456
- Issue :
- 2
- Database :
- Academic Search Index
- Journal :
- Journal of Mathematical Analysis & Applications
- Publication Type :
- Academic Journal
- Accession number :
- 124723269
- Full Text :
- https://doi.org/10.1016/j.jmaa.2017.07.044