Spectral gap estimates for the biharmonic operator on submanifolds of negatively curved spaces

Abstract

In this paper, we firstly establish two general functional inequalities on bounded domains of Riemannian manifolds carrying a special kind of function. Using this general inequalities and the comparison technique, we thereby obtain lower bound estimates for the first eigenvalues of the biharmonic operators on domains of submanifolds with controlled mean curvature and under various extrinsic curvature conditions. Meanwhile, we give some higher-order estimates concerning these problems.