Emily Proctor - Academia.edu (original) (raw)

Papers by Emily Proctor

We prove the existence of nontrivial multiparameter isospectral deformations of metrics on the cl... more We prove the existence of nontrivial multiparameter isospectral deformations of metrics on the classical compact simple Lie groups SO(n) (n = 9, n ≥ 11), Spin(n) (n = 9, n ≥ 11), SU (n) (n ≥ 7), and Sp(n) (n ≥ 5). The proof breaks into three main steps. First we outline a method devised by Schueth for constructing metrics on these groups from linear maps. Isospectrality and equivalence of linear maps are defined and we invoke a theorem by Schueth which states that isospectral linear maps give rise to isospectral metrics. Next we prove the existence of multidimensional families of linear maps such that within each family the maps are pairwise isospectral and pairwise not equivalent. Finally, we prove that generically, if F is a family of metrics arising from a collection of pairwise nonequivalent linear maps, then for any metric g contained in F there are at most finitely many metrics in F which are isometric to g.

The Michigan Mathematical Journal, 2005

Annals of Global Analysis and Geometry, 2012

ABSTRACT We show that any collection of n-dimensional orbifolds with sectional curvature and volu... more ABSTRACT We show that any collection of n-dimensional orbifolds with sectional curvature and volume uniformly bounded below, diameter bounded above, and with only isolated singular points contains orbifolds of only finitely many orbifold homeomorphism types. This is a generalization to the orbifold category of a similar result for manifolds proven by Grove, Petersen, and Wu. It follows that any Laplace isospectral collection of orbifolds with sectional curvature uniformly bounded below and having only isolated singular points also contains only finitely many orbifold homeomorphism types. The main steps of the argument are to show that any sequence from the collection has subsequence that converges to an orbifold, and then to show that the homeomorphism between the underlying spaces of the limit orbifold and an orbifold from the subsequence that is guaranteed by Perelman's stability theorem must preserve orbifold structure.

We construct a Laplace isospectral deformation of metrics on an orbifold quotient of a nilmanifol... more We construct a Laplace isospectral deformation of metrics on an orbifold quotient of a nilmanifold. Each orbifold in the deformation contains singular points with order two isotropy. Isospectrality is obtained by modifying a generalization of Sunada's Theorem due to DeTurck and Gordon.

We show that a Laplace isospectral family of two dimensional Riemannian orbifolds, sharing a lowe... more We show that a Laplace isospectral family of two dimensional Riemannian orbifolds, sharing a lower bound on sectional curvature, contains orbifolds of only a finite number of orbifold category diffeomorphism types. We also show that orbifolds of only finitely many orbifold diffeomorphism types may arise in any collection of 2-orbifolds satisfying lower bounds on sectional curvature and volume, and an upper bound on diameter. An argument converting spectral data to geometric bounds shows that the first result is a consequence of the second.

Differential Geometry and its Applications, 2010

We show that a Laplace isospectral family of two dimensional Riemannian orbifolds, sharing a lowe... more We show that a Laplace isospectral family of two dimensional Riemannian orbifolds, sharing a lower bound on sectional curvature, contains orbifolds of only a finite number of orbifold category diffeomorphism types. We also show that orbifolds of only finitely many orbifold diffeomorphism types may arise in any collection of 2-orbifolds satisfying lower bounds on sectional curvature and volume, and an upper bound on diameter. An argument converting spectral data to geometric bounds shows that the first result is a consequence of the second.

Canadian Mathematical Bulletin, 2010

We construct a Laplace isospectral deformation of metrics on an orbifold quotient of a nilmanifol... more We construct a Laplace isospectral deformation of metrics on an orbifold quotient of a nilmanifold. Each orbifold in the deformation contains singular points with order two isotropy. Isospectrality is obtained by modifying a generalization of Sunada's Theorem due to DeTurck and Gordon.

Transactions of the American Mathematical Society, 2013

We introduce the Γ-extension of the spectrum of the Laplacian of a Riemannian orbifold, where Γ i... more We introduce the Γ-extension of the spectrum of the Laplacian of a Riemannian orbifold, where Γ is a finitely generated discrete group. This extension, called the Γ-spectrum, is the union of the Laplace spectra of the Γ-sectors of the orbifold, and hence constitutes a Riemannian invariant that is directly related to the singular set of the orbifold. We compare the Γspectra of known examples of isospectral pairs and families of orbifolds and demonstrate that in many cases, isospectral orbifolds need not be Γ-isospectral. We additionally prove a version of Sunada's theorem that allows us to construct pairs of orbifolds that are Γ-isospectral for any choice of Γ.

We prove the existence of nontrivial multiparameter isospectral deformations of metrics on the cl... more We prove the existence of nontrivial multiparameter isospectral deformations of metrics on the classical compact simple Lie groups SO(n) (n = 9, n ≥ 11), Spin(n) (n = 9, n ≥ 11), SU (n) (n ≥ 7), and Sp(n) (n ≥ 5). The proof breaks into three main steps. First we outline a method devised by Schueth for constructing metrics on these groups from linear maps. Isospectrality and equivalence of linear maps are defined and we invoke a theorem by Schueth which states that isospectral linear maps give rise to isospectral metrics. Next we prove the existence of multidimensional families of linear maps such that within each family the maps are pairwise isospectral and pairwise not equivalent. Finally, we prove that generically, if F is a family of metrics arising from a collection of pairwise nonequivalent linear maps, then for any metric g contained in F there are at most finitely many metrics in F which are isometric to g.

The Michigan Mathematical Journal, 2005

Annals of Global Analysis and Geometry, 2012

ABSTRACT We show that any collection of n-dimensional orbifolds with sectional curvature and volu... more ABSTRACT We show that any collection of n-dimensional orbifolds with sectional curvature and volume uniformly bounded below, diameter bounded above, and with only isolated singular points contains orbifolds of only finitely many orbifold homeomorphism types. This is a generalization to the orbifold category of a similar result for manifolds proven by Grove, Petersen, and Wu. It follows that any Laplace isospectral collection of orbifolds with sectional curvature uniformly bounded below and having only isolated singular points also contains only finitely many orbifold homeomorphism types. The main steps of the argument are to show that any sequence from the collection has subsequence that converges to an orbifold, and then to show that the homeomorphism between the underlying spaces of the limit orbifold and an orbifold from the subsequence that is guaranteed by Perelman's stability theorem must preserve orbifold structure.

We construct a Laplace isospectral deformation of metrics on an orbifold quotient of a nilmanifol... more We construct a Laplace isospectral deformation of metrics on an orbifold quotient of a nilmanifold. Each orbifold in the deformation contains singular points with order two isotropy. Isospectrality is obtained by modifying a generalization of Sunada's Theorem due to DeTurck and Gordon.

We show that a Laplace isospectral family of two dimensional Riemannian orbifolds, sharing a lowe... more We show that a Laplace isospectral family of two dimensional Riemannian orbifolds, sharing a lower bound on sectional curvature, contains orbifolds of only a finite number of orbifold category diffeomorphism types. We also show that orbifolds of only finitely many orbifold diffeomorphism types may arise in any collection of 2-orbifolds satisfying lower bounds on sectional curvature and volume, and an upper bound on diameter. An argument converting spectral data to geometric bounds shows that the first result is a consequence of the second.

Differential Geometry and its Applications, 2010

We show that a Laplace isospectral family of two dimensional Riemannian orbifolds, sharing a lowe... more We show that a Laplace isospectral family of two dimensional Riemannian orbifolds, sharing a lower bound on sectional curvature, contains orbifolds of only a finite number of orbifold category diffeomorphism types. We also show that orbifolds of only finitely many orbifold diffeomorphism types may arise in any collection of 2-orbifolds satisfying lower bounds on sectional curvature and volume, and an upper bound on diameter. An argument converting spectral data to geometric bounds shows that the first result is a consequence of the second.

Canadian Mathematical Bulletin, 2010

We construct a Laplace isospectral deformation of metrics on an orbifold quotient of a nilmanifol... more We construct a Laplace isospectral deformation of metrics on an orbifold quotient of a nilmanifold. Each orbifold in the deformation contains singular points with order two isotropy. Isospectrality is obtained by modifying a generalization of Sunada's Theorem due to DeTurck and Gordon.

Transactions of the American Mathematical Society, 2013

We introduce the Γ-extension of the spectrum of the Laplacian of a Riemannian orbifold, where Γ i... more We introduce the Γ-extension of the spectrum of the Laplacian of a Riemannian orbifold, where Γ is a finitely generated discrete group. This extension, called the Γ-spectrum, is the union of the Laplace spectra of the Γ-sectors of the orbifold, and hence constitutes a Riemannian invariant that is directly related to the singular set of the orbifold. We compare the Γspectra of known examples of isospectral pairs and families of orbifolds and demonstrate that in many cases, isospectral orbifolds need not be Γ-isospectral. We additionally prove a version of Sunada's theorem that allows us to construct pairs of orbifolds that are Γ-isospectral for any choice of Γ.