arXiv subjects

Craig Sutton

Publications and source records attributed to Craig Sutton.

3 recordsLinked to original sources

arXiv2022

Spectral multiplicity and nodal sets for generic torus-invariant metrics

Let a torus $T$ act freely on a closed manifold $M$ of dimension at least two. We demonstrate that, for a generic $T$-invariant Riemannian metric $g$ on $M$, each real $\Delta_g$-eigenspace is an irreducible real representation of $T$ and, therefore, has dimension at most two. We also show that, for the generic $T$-invariant metric on $M$, if $u$ is a non-invariant real-valued $\Delta_g$-eigenfunction that vanishes on some $T$-orbit, then the nodal set of $u$ is a connected smooth hypersurface whose complement has exactly two connected components.

math.DG↗

arXiv2019

Geometric structures and the Laplace spectrum, part II

We continue our exploration of the extent to which the spectrum encodes the local geometry of a locally homogeneous three-manifold and find that if $(M,g)$ and $(N,h)$ are a pair of locally homogeneous, locally non-isometric isospectral three-manifolds, where $M$ is an elliptic three-manifold, then $(1)$ $N$ is also an elliptic three-manifold, $(2)$ $M$ and $N$ have fundamental groups of different orders, $(3)$ $(M,g)$ and $(N,h)$ both have non-degenerate Ricci tensors and $(4)$ the metrics $g$ and $h$ are sufficiently far from a metric of constant sectional curvature. We are unaware of any such isospectral pair and such a pair could not arise via the classical Sunada method. As part of the proof, we provide an explicit description of the isometry group of a compact simple Lie group equipped with a left-invariant metric---improving upon the results of Ochiai-Takahashi and Onishchik---which we use to classify the locally homogeneous metrics on an elliptic three-manifold $\Gamma \backslash S^3$ and we determine that any collection of isospectral locally homogeneous metrics on an elliptic three-manifold consists of at most two isometry classes that are necessarily locally isometric. In particular, the left-invariant metrics on $\operatorname{SO}(3)$ (respectively, $S^3$) can be mutually distinguished via their spectra. The previous statement has the following interpretation in terms of physical chemistry: the moments of inertia of a molecule can be recovered from its rotational spectrum.

math.DG↗

arXiv2019

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.'

math.DG↗