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

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:456

Article

Du, Feng²  Mao, Jing¹ 

[1] Hubei Univ, Fac Math & Stat, Key Lab Appl Math Hubei Prov, Wuhan 430062, Hubei, Peoples R China

[2] Jingchu Univ Technol, Sch Math & Phys Sci, Jingmen 448000, Peoples R China

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

DOI  :  10.1016/j.jmaa.2017.07.044

来源: Elsevier

【 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.

【 授权许可】

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