Estimates for the first eigenvalue of the drifting Laplace, and the p-Laplace operators on submanifolds with bounded mean curvature in the hyperbolic space

[摘要] 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 vertical bar vertical bar H vertical bar vertical bar satisfies vertical bar vertical bar H vertical bar vertical bar <= alpha < n-1, then we prove that the first eigenvalue lambda(1,p)(M) of the p-Laplacian Delta(p) on M satisfies lambda(1,p)(M) >= (n-1-alpha/p)(p), 1 < p < infinity, with equality achieved when M is totally geodesic and p = 2; let (M, g, e(-phi) dv(g)) be an n-dimensional complete noncompact smooth metric measure space with M being a submanifold in a hyperbolic space, and vertical bar vertical bar H vertical bar vertical bar <= alpha < n-1, vertical bar vertical bar del(phi)vertical bar vertical bar <= C with del the gradient operator on M, then we show that the first eigenvalue (M) of the weighted Laplacian del(phi) on M satisfies lambda(1,p)(M) >= (n-1-alpha-c/4)(2), with equality attained when M is totally geodesic and phi = constant. (C) 2017 Elsevier Inc. All rights reserved.

[发布日期] 2017-12-15

[关键词] Eigenvalues;Laplacian;Drifting Laplacian;p-Laplacian;Smooth metric measure spaces