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

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.