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

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- by David D. Bleecker PDF
- Trans. Amer. Math. Soc.
**275**(1983), 409-416 Request permission

## 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.## References

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## Additional Information

- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**275**(1983), 409-416 - MSC: Primary 53C20; Secondary 58G30
- DOI: https://doi.org/10.1090/S0002-9947-1983-0678360-0
- MathSciNet review: 678360