Spectral bounds on closed hyperbolic 3-manifolds (Q2840170)

In this paper the author works out a quantitative relation between the geometry and the topology on the one side, and the spectral theory on the other side, for closed hyperbolic three-manifolds. More specifically, for such a manifold \(M\) let \(0<\lambda_1\leq\dots\leq\lambda_k\) be the \(k\) first positive eigenvalues of the Laplace operator on functions on \(M\), and for \(\varepsilon,c>0\) say that a closed hyperbolic three-manifold \(M\) satisfies to condition (\(\ast\)) if its injectivity radius is bounded below by \(\varepsilon\) and its fundamental group \(\pi_1(M)\) is of rank no greater than \(c\) (i.e. can be generated by less than \(c\) elements). She proves that for any \(k\geq 1\) there is a constant \(\Omega=\Omega(\varepsilon,c,k)\) so that for any closed hyperbolic three-manifold satisfying to (\(\ast\)) the bounds NEWLINE\[NEWLINE \Omega^{-1} (\mathrm{vol} M)^{-2} \leq \lambda_k \leq \Omega (\mathrm{vol} M)^{-2} NEWLINE\]NEWLINE hold. The lower bound is due to \textit{R. Schoen} [J. Differ. Geom. 17, 233--238 (1982; Zbl 0516.53048)]; it does not in fact depend on any of the two conditions in (\(\ast\)), in contrast with the upper bound which depends on both. As the author notes, the upper bound implies an upper bound for the volume of a manifold for which (\(\ast\)) holds and which further has \(\lambda_k\geq\delta>0\); in particular the number of such manifolds is finite.NEWLINENEWLINEThis last result for \(k=1\) was already known thanks to the work of \textit{I. Biringer} and \textit{J. Souto} [J. Lond. Math. Soc., II. Ser. 84, No. 1, 227--242 (2011; Zbl 1233.57008)], and to prove the upper bound the present paper uses a quantitative refinement of their method which we will now describe. It is based on a description of the large-scale geometry of a manifold satisfying (\(\ast\)) which is of independent interest: such a manifold will be a union of a bounded number of pieces with prescribed coarse geometry, and the complement to those will be a union of ``product regions'', each of them homeomorphic to some \(S\times\mathbb{R}\) where \(S\) is a closed surface. A coarse description of the geometry of these pieces away from their boundary is then sufficient to construct a function supported in this region with small Rayleigh quotient -- thus getting an upper bound on \(\lambda_1\) depending on the volume. The main part of the author's argument is to refine substantially Biringer and Souto's description of the geometry of the product regions -- this she states in Theorem 10, the proof of which takes most of the paper -- which allows her to construct \(k\) linearly independent functions with small enough Rayleigh quotients to deduce the upper bound on \(\lambda_k\).