Hironori Kumura

Limiting absorption principle on manifolds having ends with various measure growth rate limits

The purpose of this paper is to study the property of the resolvent of the Laplace-Beltrami operator on a noncompact complete Riemannian manifold with various ends each of which has a different limit of the growth rate of the Riemannian measure at infinity, in particular, focusing on the limiting absorption principle. As a result, we will obtain the absolute continuity of the Laplace-Beltrami operator.

On the Essential Spectrum of the Laplacian and Vague Convergence of the Curvature at Infinity

We shall prove that under some volume growth condition, the essential spectrum of the Laplacian contains the interval [(n − 1)2 K/4, ∞) if an n-dimensional Rieman-nian manifold has an end and the average of the part of the Ricci curvature on the end which lies below a nonpositive constant (n − 1)K converges to zero at infinity.