EIGENVALUES OF THE LAPLACIAN ON RIEMANNIAN MANIFOLDS

Bounds of Eigenvalues on Riemannian Manifolds

In this paper, we first give a short review of the eigenvalue estimates of Laplace operator and Schrödinger operators. Then we discuss the evolution of eigenvalues along the Ricci flow, and two new bounds of the first eigenvalue using gradient estimates. 2000 Mathematics Subject Classification: 58J50, 35P15, 53C21.

Eigenvalues and Capacities of Riemannian Manifolds

This paper is concerned with eigenvalues of the biharmonic operators and the buckling eigenvalue for complete Riemannian manifolds. We are mostly concerned with relating bounds for these eigenvalues to the behavior of the ends of the manifold. Let M be a complete Riemannian manifold. M is called parabolic if every non-positive subharmonic function on M reduces to a constant. By an end E of M we...

Universal Bounds for Eigenvalues of Schrödinger Operator on Riemannian Manifolds

Abstract. In this paper we consider eigenvalues of Schrödinger operator with a weight on compact Riemannian manifolds with boundary (possibly empty) and prove a general inequality for them. By using this inequality, we study eigenvalues of Schrödinger operator with a weight on compact domains in a unit sphere, a complex projective space and a minimal submanifold in a Euclidean space. We also st...

Higher eigenvalues and isoperimetric inequalities on Riemannian manifolds and graphs

5 Analysis on weighted graphs 23 5.1 Measures on graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.2 Discrete Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.3 Green’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.4 Integration versus Summation . . . . . . . . . . . . . . . . . . . . . . ...

Eigenvalues of the Laplacian and Invariants of Manifolds