The smallest positive eigenvalue of fibered hyperbolic 3‐manifolds

Abstract

We study the smallest positive eigenvalue λ1false(Mfalse) of the Laplace–Beltrami operator on a closed hyperbolic 3‐manifold M which fibers over the circle, with fiber a closed surface of genus g⩾2. We show the existence of a constant C>0 only depending on g so that λ1false(Mfalse)∈false[C−1/ vol false(Mfalse)2,Clog vol (M)/ vol false(Mfalse)22g−2/(22g−2−1)false] and that this estimate is essentially sharp. We show that if M is typical or random, then we have λ1false(Mfalse)∈false[C−1/ vol false(Mfalse)2,C/ vol false(Mfalse)2false]. This rests on a result of independent interest about reccurence properties of axes of random pseudo‐Anosov elements.

Small eigenvalues and thick-thin decomposition in negative curvature

“…for any closed hyperbolic 3-manifold whose injectivity radius is bounded from below by δ and such that the rank of the fundamental group π 1 (M ) of M is at most r. This rank is defined to be the minimal number of generators of π 1 (M ), and it is bounded from above by a fixed multiple of the volume ([12, Theorem 1.10]). A similar statement also holds true for random 3-dimensional hyperbolic mapping tori [1] and for random hyperbolic 3manifolds of fixed Heegaard genus [14].…”

“…The small part of the spectrum of the Laplace operator ∆ acting on functions on a closed oriented hyperbolic surface S is quite well understood. Namely, if g denotes the genus of S then the 2g − 3-th eigenvalue λ 2g−3 (S) can be arbitrarily small [4], while λ 2g−2 (S) > 1 4 [18]. By the Gauss-Bonnet theorem, the volume of S equals 2π(2g − 2), so these results relate the small part of the spectrum of S to its volume.…”

Volumes of random 3‐manifolds

“…H 2 satisfying the three requirements (1) Topologically, H 1 and H 2 are homeomorphic to genus g handlebodies, while Q, Ω 1 and Ω 2 are homeomorphic to Σ × [0, 1]. (2) Geometrically, ρ has negative curvature sec ∈ (−1 − ǫ, −1 + ǫ), but outside the region Ω = Ω 1 ∪ Ω 2 the metric is purely hyperbolic.…”

“…The idea is the following: We can obtain a hyperbolic metric on M f by a hyperbolic cone manifold deformation from a finite volume metric on a drilled manifold M which has the following form: Let Σ × [1,4] be a tubular neighbourhood of Σ ⊂ M f . We consider 3-manifolds diffeomorphic to…”

“…In the paper, we will make this heuristic picture more accurate. Our three main ingredients are the model manifold, bridging between the geometry of the Teichmüller space and the internal geometry of quasi‐Fuchsian manifolds [12, 38], a recurrence property for random walks [1] and the method of natural maps from Besson–Courtois–Gallot [4]. They correspond, respectively, to Proposition 3.10, Theorem 4.3 and Theorem 3.11.…”