Eigenvalues Of The Bi-Xin-Laplacian On Complete Riemannian Manifolds

The clamped plate problem stems from the vibration of a clamped plate in the elastic mechanics and Xin-Laplacian is an important elliptic operator for understanding the geometric structure of translators of mean curvature flow(MCF for short).In this thesis,we investigate the clamped plate problem of the bi-Xin-Laplacian on Riemannian manifolds isometrically immersed in the Euclidean space.On one hand,we obtain some eigenvalue inequalities of bi-Xin-Laplacian on some important Riemannian manifolds admitting some special functions.Let us emphasize that,this class of manifolds contains some interesting examples: Cartan-Hadamard manifolds,some types of warp product manifolds and homogenous spaces.On the other hand,we also consider the eigenvalue of the bi-Xin-Laplacian on the cylinders and obtain an eigenvalue inequality.In particular,we can give an estimate for the lower order eigenvalues on the cylinders.