Geometric structures and the Laplace spectrum

Inspired by the role geometric structures play in our understanding of surfaces and three-manifolds, and Berger's observation that a surface of constant sectional curvature is determined up to local isometry by its Laplace spectrum, we explore the extent to which compact locally homogeneous three-manifolds are characterized up to local isometry by their spectra. We observe that there are eight `metrically maximal' three-dimensional geometries on which all compact locally homogeneous three-manifolds are modeled and we demonstrate that for five of these geometries the associated compact locally homogeneous three-manifolds are determined up to local isometry by their spectra within the universe of locally homogeneous three-manifolds. Specifically, we show that among compact locally homogeneous three-manifolds, a Riemannian three-manifold is determined up to local isometry if its universal Riemannian cover is isometric to (1) a symmetric space, (2) $\mathbb{R}^2 \rtimes \mathbb{R}$ endowed with a left-invariant metric, (3) $\operatorname{Nil}$ endowed with a left-invariant metric, or (4) $S^3$ endowed with a left-invariant metric sufficiently close to a metric of constant sectional curvature. We then deduce that three-dimensional Riemannian nilmanifolds and locally symmetric spaces with universal Riemannian cover $\mathbb{S}^2 \times \mathbb{E}$ are uniquely characterized by their spectra among compact locally homogeneous three-manifolds. Finally, within the collection of closed manifolds covered by $\operatorname{Sol}$ equipped with a left-invariant metric, local geometry is `audible.'