Boundedness of Laplacian eigenfunctions on manifolds of infinite volume

Abstract

In a Hadamard manifold M , it is proved that if u is a λ-eigenfunction of the Laplacian that belongs to L p (M ) for some p ≥ 2, then u is bounded and u ∞ ≤ C u p , where C depends only on p, λ and on the dimension of M . This result is obtained in the more general context of a complete Riemannian manifold endowed with an isoperimetric function H satisfying some integrability condition. In this case, the constant C depends on p, λ and H.

A characterization related to Schrödinger equations on Riemannian manifolds

“…Later, Shen [She03] characterized the discretness of the spectrum by using the basic length scale function and the effective potential function. For further recent studies in this topic, we invite the reader to consult the papers [CM11,CM13,BKT16].…”