arXiv subjects

Carolyn Gordon

Publications and source records attributed to Carolyn Gordon.

13 recordsLinked to original sources

arXiv2026

The Steklov spectrum of convex polygonal domains II: investigating spectral determination

The extent to which the geometry of an object is determined by some associated spectral data is a longstanding problem. We investigate this problem in the context of the Steklov spectrum, focusing on convex polygons. We prove that almost all triangles are uniquely determined by their Steklov spectra within the class of all triangles; further results depending on the types of angles in the triangles are given. We examine three special classes of convex quadrilaterals--rectangles, parallelograms, and kites--and obtain results ranging from unique spectral determination to determination up to three possibilities. For regular $n$-gons, we are again able to prove spectral determination within certain classes of polygons. More generally, we investigate the extent to which the Steklov spectrum distinguishes convex polygons from simply-connected domains with smooth boundary; that is, does the Steklov spectrum detect corners? We prove that triangles and quadrilaterals are spectrally distinguished from such smoothly bounded domains; moreover, we show that having the same Steklov spectrum as such a domain imposes substantial restrictions on the edge lengths of higher-order $n$-gons. Throughout, our main tool is the characteristic polynomial developed in works by Stanislav Krymski, Michael Levitin, Leonid Parnovski, Iosif Polterovich, and David A. Sher.

math.DG↗

arXiv2024

The Steklov spectrum of convex polygonal domains I: spectral finiteness

We explore the Steklov eigenvalue problem on convex polygons, focusing mainly on the inverse Steklov problem. Our primary finding reveals that, for almost all convex polygonal domains, there exist at most finitely many non-congruent domains with the same Steklov spectrum. Moreover, we obtain explicit upper bounds for the maximum number of mutually Steklov isospectral non-congruent polygonal domains. Along the way, we obtain isoperimetric bounds for the Steklov eigenvalues of a convex polygon in terms of the minimal interior angle of the polygon.

math.SP↗

arXiv2024

Do the Hodge spectra distinguish orbifolds from manifolds? Part 2

In \cite{GGKM-SSS} we examined the relationship between the singular set of a compact Riemannian orbifold and the spectrum of the Hodge Laplacian on $p$-forms by computing the heat invariants associated to the $p$-spectrum. We showed that the heat invariants of the $0$-spectrum together with those of the $1$-spectrum for the corresponding Hodge Laplacians are sufficient to distinguish orbifolds from manifolds as long as the singular sets have codimension $\le 3.$ This is enough to distinguish orbifolds from manifolds for dimension $\le 3.$ Here we give both positive and negative inverse spectral results for the individual $p$-spectra considered separately. For example, we give conditions on the codimension of the singular set which guarantee that the volume of the singular set is determined, and in many cases we show by providing counterexamples that the conditions are sharp.

math.DG↗

arXiv2023

Some recent developments on the Steklov eigenvalue problem

The Steklov eigenvalue problem, first introduced over 125 years ago, has seen a surge of interest in the past few decades. This article is a tour of some of the recent developments linking the Steklov eigenvalues and eigenfunctions of compact Riemannian manifolds to the geometry of the manifolds. Topics include isoperimetric-type upper and lower bounds on Steklov eigenvalues (first in the case of surfaces and then in higher dimensions), stability and instability of eigenvalues under deformations of the Riemannian metric, optimisation of eigenvalues and connections to free boundary minimal surfaces in balls, inverse problems and isospectrality, discretisation, and the geometry of eigenfunctions. We begin with background material and motivating examples for readers that are new to the subject. Throughout the tour, we frequently compare and contrast the behavior of the Steklov spectrum with that of the Laplace spectrum. We include many open problems in this rapidly expanding area.

math.SP↗

arXiv2023

Infinitesimal maximal symmetry and Ricci soliton solvmanifolds

This work addresses the questions: (i) Among all left-invariant Riemannian metrics on a given Lie group, is there any whose isometry group or isometry algebra contain that of all others? (ii) Do expanding left-invariant Ricci solitons exhibit such maximal symmetry? Question (i) is addressed both for semisimple and for solvable Lie groups. Building on previous work of the authors on Einstein metrics, a complete answer is given to (ii): expanding homogeneous Ricci solitons have maximal isometry algebras although not always maximal isometry groups. As a consequence of the tools developed to address these questions, partial results of Boehm, Lafuente, and Lauret are extended to show that left-invariant Ricci solitons on solvable Lie groups are unique up to scaling and isometry.

math.DG↗

arXiv2021

Do the Hodge spectra distinguish orbifolds from manifolds? Part 1

We examine the relationship between the singular set of a compact Riemannian orbifold and the spectrum of the Hodge Laplacian on $p$-forms by computing the heat invariants associated to the $p$-spectrum. We show that the heat invariants of the $0$-spectrum together with those of the $1$-spectrum for the corresponding Hodge Laplacians are sufficient to distinguish orbifolds with singularities from manifolds as long as the singular sets have codimension $\le 3.$ This is enough to distinguish orbifolds from manifolds for dimension $\le 3.$

math.DG↗

arXiv2018

Robin and Steklov isospectral manifolds

We use two of the most fruitful methods for constructing isospectral manifolds, the Sunada method and the torus action method, to construct manifolds whose Dirichlet-to-Neumann operators are isospectral at all frequencies. The manifolds are also isospectral for the Robin boundary value problem for all choices of Robin parameter. As in the sloshing problem, we can also impose mixed Dirichlet-Neumann conditions on parts of the boundary. Among the examples we exhibit are Steklov isospectral flat surfaces with boundary, planar domains with isospectral sloshing problems, and Steklov isospectral metrics on balls of any dimension greater than 5. In particular, the latter are the first examples of Steklov isospectral manifolds of dimension greater than 2 that have connected boundaries.

math.DG↗

arXiv2018

Transplantation and isogeny of intermediate Jacobians of compact Kähler manifolds

We give a general method for constructing compact Kähler manifolds $X_1$ and $X_2$ whose intermediate Jacobians $J^k(X_1)$ and $J^k(X_2)$ are isogenous for each $k$, and we exhibit some examples. The method is based upon the algebraic transplantation formalism arising from Sunada's technique for constructing pairs of compact Riemannian manifolds whose Laplace spectra are the same. We also show that the method produces compact Riemannian manifolds whose Lazzeri Jacobians are isogenous.

math.AG↗

arXiv2011

Classical Equivalence and Quantum Equivalence of Magnetic Fields on Flat Tori

Let M be a real 2m-torus equipped with a translation-invariant metric h and a translation-invariant symplectic form w; the latter we interpret as a magnetic field on M. The Hamiltonian flow of half the norm-squared function induced by h on T^*M (the "kinetic energy") with respect to the twisted symplectic form w_{T^*M}+ π^*w describes the trajectories of a particle moving on M under the influence of the magnetic field w. If [w] is an integral cohomology class, then we can study the geometric quantization of the symplectic manifold (T^*M,w_{T^*M}+π^*w) with the kinetic energy Hamiltonian. We say that the quantizations of two such tori (M_1,h_1,w_1) and (M_2,h_2,w_2) are quantum equivalent if their quantum spectra, i.e., the spectra of the associated quantum Hamiltonian operators, coincide; these quantum Hamiltonian operators are proportional to the h_j-induced bundle Laplacians on powers of the Hermitian line bundle on M with Chern class [w]. In this paper, we construct continuous families {(M,h_t)}_t of mutually nonisospectral flat tori (M,h_t), each endowed with a translation-invariant symplectic structure w, such that the associated classical Hamiltonian systems are pairwise equivalent. If w represents an integer cohomology class, then the (M,h_t,w) also have the same quantum spectra. We show moreover that for any translation-invariant metric h and any translation-invariant symplectic structure w on M that represents an integer cohomology class, the associated quantum spectrum determines whether (M,h,w) is Kaehler, and that all translation-invariant Kaehler structures (h,w) of given volume on M have the same quantum spectra. Finally, we construct pairs of magnetic fields (M,h,w_1), (M,h,w_2) having the same quantum spectra but nonsymplectomorphic classical phase spaces. In some of these examples the pairs consist of Kaehler manifolds.

math.DG↗

arXiv2010

Quantum Equivalent Magnetic Fields that Are Not Classically Equivalent

We construct pairs of compact Kähler-Einstein manifolds $(M_i,g_i,ω_i)$ ($i=1,2)$ of complex dimension $n$ with the following properties: The canonical line bundle $L_i=\bigwedge^n T^*M_i$ has Chern class $[ω_i/2π]$, and for each integer $k$ the tensor powers $L_1^{\otimes k}$ and $L_2^{\otimes k}$ are isospectral for the bundle Laplacian associated with the canonical connection, while $M_1$ and $M_2$ -- and hence $T^*M_1$ and $T^*M_2$ -- are not homeomorphic. In the context of geometric quantization, we interpret these examples as magnetic fields which are quantum equivalent but not classically equivalent. Moreover, we construct many examples of line bundles $L$, pairs of potentials $Q_1$, $Q_2$ on the base manifold, and pairs of connections $\nabla_1$, $\nabla_2$ on $L$ such that for each integer $k$ the associated Schrödinger operators on $L^{\otimes k}$ are isospectral.

math.DG↗

arXiv2002

Continuous Families of Isophasal Scattering Manifolds

We construct continuous families of scattering manifolds with the same scattering phase. The manifolds are compactly supported metric perturbations of Euclidean $\mathbf{R}^{n}$ for $n\geq8$. The metric perturbation may have arbitrarily small support.

math.DG↗

arXiv1995

Geodesic Conjugacy in two-step nilmanifolds

Two Riemannian manifolds are said to have $C^k$-conjugate geodesic flows if there exist an $C^k$ diffeomorphism between their unit tangent bundles which intertwines the geodesic flows. We obtain a number of rigidity results for the geodesic flows on compact 2-step Riemannian nilmanifolds: For generic 2-step nilmanifolds the geodesic flow is $C^2$ rigid. For special classes of 2-step nilmanifolds, we show that the geodesic flow is $C^0$ or $C^2$ rigid. In particular, there exist continuous families of 2-step nilmanifolds whose Laplacians are isospectral but whose geodesic flows are not $C^0$ conjugate.

math.DG↗

arXiv1992

One cannot hear the shape of a drum

We use an extension of Sunada's theorem to construct a nonisometric pair of isospectral simply connected domains in the Euclidean plane, thus answering negatively Kac's question, ``can one hear the shape of a drum?'' In order to construct simply connected examples, we exploit the observation that an orbifold whose underlying space is a simply connected manifold with boundary need not be simply connected as an orbifold.

math.DG↗