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

Abstract

Let ( P , g ) (P,g) be a compact, connected, C ∞ {C^\infty } Riemannian ( n + 1 ) (n + 1) -manifold ( n ⩾ 1 ) (n \geqslant 1) with a unit Killing vector field with dual 1 1 -form η \eta . For t > 0 t > 0 , let g t = t − 1 g + ( t n − t − 1 ) η ⊗ η {g_{t}} = {t^{ - 1}}g + (t^{n}-t^{-1})\eta \otimes \eta , a family of metrics of fixed volume element on P P . Let λ 1 ( t ) {\lambda _1}(t) be the first nonzero eigenvalue of the Laplace operator on C ∞ ( P ) {C^\infty }(P) of the metric g t {g_t} . We prove that if d η d\eta is nowhere zero, then λ 1 ( t ) → ∞ {\lambda _1}(t) \to \infty as t → ∞ 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 λ 1 {\lambda _1} is unbounded on the space of fixed-volume metrics.