First Eigenvalue Estimates Of The Weighted Laplace And The P-Laplace Operators On Submanifold

The eigenvalue problem is a hot research field in recent years.Its research theory is closely related to Riemannian geometry,submanifold geometry and partial differen-tial equations,etc.In this paper,if φ:M→Q is an isometric immersion with locally mean curvature bounded,we can obtain the lower bounds of λ1,φ*(Ω)and λ1,p(Ω)for any connected component Ω of Φ-1(BQ(q,r))under the different sectional curvatures.Furthermore,if M is noncompact with bounded mean curvature(stronger than the lo-cally bounded mean curvature),and the sectional curvature is non-positive,then we can show that λ1,φ*(Ω)and λ1,p(Ω)have positive lower bounds.