Volume 5, Issue 4
The Problem of Eigenvalue on Noncompact Complete Riemannian Manifold

Li Jiayu

DOI:

J. Part. Diff. Eq.,5(1992),pp.87-95

Published online: 1992-05

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  • Abstract

Let M be an n-dimensional noncompact complete Riemannian manifold, "Δ" is the Laplacian of M. It is a negative selfadjoint operator in L²(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

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@Article{JPDE-5-87, author = {Li Jiayu}, title = {The Problem of Eigenvalue on Noncompact Complete Riemannian Manifold}, journal = {Journal of Partial Differential Equations}, year = {1992}, volume = {5}, number = {4}, pages = {87--95}, abstract = { Let M be an n-dimensional noncompact complete Riemannian manifold, "Δ" is the Laplacian of M. It is a negative selfadjoint operator in L²(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.}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5756.html} }
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, "Δ" is the Laplacian of M. It is a negative selfadjoint operator in L²(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.
Li Jiayu. (1970). The Problem of Eigenvalue on Noncompact Complete Riemannian Manifold. Journal of Partial Differential Equations. 5 (4). 87-95. doi:
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