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

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 λ1,p(M) of the p-Laplacian Δp on M satisfies λ1,p(M)⩾(n−1−αp)p, 1<p<∞, with equality achieved when M is totally geodesic and p=2; let (M,g,e−φdvg) 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 λ1,φ(M) of the weighted Laplacian Δφ on M satisfies λ1,φ(M)⩾(n−1−α−C)24, with equality attained when M is totally geodesic and φ=constant.