Eigenvalues and eigenfunctions of Riemannian manifolds

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...

Riemannian Manifolds with Uniformly Bounded Eigenfunctions

The standard eigenfunctions φλ = ei〈λ,x〉 on flat tori Rn/L have L-norms bounded independently of the eigenvalue. In the case of irrational flat tori, it follows that L2normalized eigenfunctions have uniformly bounded L-norms. Similar bases exist on other flat manifolds. Does this property characterize flat manifolds? We give an affirmative answer for compact Riemannian manifolds with quantum co...

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.

Local and Global Analysis of Eigenfunctions on Riemannian Manifolds

This is a survey on eigenfunctions of the Laplacian on Riemannian manifolds (mainly compact and without boundary). We discuss both local results obtained by analyzing eigenfunctions on small balls, and global results obtained by wave equation methods. Among the main topics are nodal sets, quantum limits, and L norms of global eigenfunctions. The emphasis is on the connection between the behavio...

Blowing Up at Infinity of Eigenfunctions on Riemannian Manifolds