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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

The spectrum of a Riemannian manifold with a unit Killing vector field


Author: David D. Bleecker
Journal: Trans. Amer. Math. Soc. 275 (1983), 409-416
MSC: Primary 53C20; Secondary 58G30
MathSciNet review: 678360
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Abstract: Let $ (P,g)$ be a compact, connected, $ {C^\infty }$ Riemannian $ (n + 1)$-manifold $ (n \geqslant 1)$ with a unit Killing vector field with dual $ 1$-form $ \eta $. For $ t > 0$, let $ {g_{t}} = {t^{ - 1}}g + (t^{n}-t^{-1})\eta \otimes \eta$, a family of metrics of fixed volume element on $ P$. Let $ {\lambda _1}(t)$ be the first nonzero eigenvalue of the Laplace operator on $ {C^\infty }(P)$ of the metric $ {g_t}$. We prove that if $ d\eta $ is nowhere zero, then $ {\lambda _1}(t) \to \infty$ as $ t \to \infty $. Using this construction, we find that, for every dimension greater than two, there are infinitely many topologically distinct compact manifolds for which $ {\lambda _1}$ is unbounded on the space of fixed-volume metrics.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1983-0678360-0
PII: S 0002-9947(1983)0678360-0
Keywords: Laplacian, Riemannian manifold, Killing vector field, spectrum
Article copyright: © Copyright 1983 American Mathematical Society