TY - JOUR
T1 - The Problem of Eigenvalue on Noncompact Complete Riemannian Manifold
AU - Li Jiayu
JO - Journal of Partial Differential Equations
VL - 4
SP - 87
EP - 95
PY - 1992
DA - 1992/05
SN - 5
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jpde/5756.html
KW - Laplacian
KW -  spectrum
KW -  eigenvalue
AB -  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.