The Problem of Eigenvalue on Noncompact Complete Riemannian Manifold (Journal of Partial Differential Equations, 2020)
Let M be an n-dimensional noncompact complete Riemannian manifold, "&#916" is the Laplacian of M. It is a negative selfadjoint operator in L&sup2(M). First, we give a criterion of non-existence of eigenvalue by the heat kernel. Applying the criterion yields that the Laplacian on noncompact constant curvature space form has no eigenvalue. Then, we give a geometric condition of M under which the Laplacian of M has eigenvalues. It implies that changing the metric on a compact domain of constant negative curvature space form may yield eigenvalues.
Keywords: Laplacian; spectrum; eigenvalue