67 research outputs found
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.Comment: 157 pages, 7 figures. To appear in Revista Matem\'atica Complutens
Constrained Willmore Tori and Elastic Curves in 2-Dimensional Space Forms
In this paper we consider two special classes of constrained Willmore tori in
the 3-sphere. The first class is given by the rotation of closed elastic curves
in the upper half plane - viewed as the hyperbolic plane - around the x-axis.
The second is given as the preimage of closed constrained elastic curves, i.e.,
elastic curve with enclosed area constraint, in the round 2-sphere under the
Hopf fibration. We show that all conformal types can be isometrically immersed
into S^3 as constrained Willmore (Hopf) tori and write down all constrained
elastic curves in H^2 and S^2 in terms of the Weierstrass elliptic functions.
Further, we determine the closing condition for the curves and compute the
Willmore energy and the conformal type of the resulting tori.Comment: 23 pages, 2 figure
Spectral geometry of the Steklov problem
The Steklov problem is an eigenvalue problem with the spectral parameter in
the boundary conditions, which has various applications. Its spectrum coincides
with that of the Dirichlet-to-Neumann operator. Over the past years, there has
been a growing interest in the Steklov problem from the viewpoint of spectral
geometry. While this problem shares some common properties with its more
familiar Dirichlet and Neumann cousins, its eigenvalues and eigenfunctions have
a number of distinctive geometric features, which makes the subject especially
appealing. In this survey we discuss some recent advances and open questions,
particularly in the study of spectral asymptotics, spectral invariants,
eigenvalue estimates, and nodal geometry.Comment: 26 pages, 7 figure
Spectral preorder and perturbations of discrete weighted graphs
In this article, we introduce a geometric and a spectral preorder relation on
the class of weighted graphs with a magnetic potential. The first preorder is
expressed through the existence of a graph homomorphism respecting the magnetic
potential and fulfilling certain inequalities for the weights. The second
preorder refers to the spectrum of the associated Laplacian of the magnetic
weighted graph. These relations give a quantitative control of the effect of
elementary and composite perturbations of the graph (deleting edges,
contracting vertices, etc.) on the spectrum of the corresponding Laplacians,
generalising interlacing of eigenvalues.
We give several applications of the preorders: we show how to classify graphs
according to these preorders and we prove the stability of certain eigenvalues
in graphs with a maximal d-clique. Moreover, we show the monotonicity of the
eigenvalues when passing to spanning subgraphs and the monotonicity of magnetic
Cheeger constants with respect to the geometric preorder. Finally, we prove a
refined procedure to detect spectral gaps in the spectrum of an infinite
covering graph.Comment: 26 pages; 8 figure
Spectral geometry of the Steklov problem on orbifolds
We consider how the geometry and topology of a compact -dimensional Riemannian orbifold with boundary relates to its Steklov spectrum. In two dimensions, motivated by work of A. Girouard, L. Parnovski, I. Polterovich and D. Sher in the manifold setting, we compute the precise asymptotics of the Steklov spectrum in terms of only boundary data. As a consequence, we prove that the Steklov spectrum detects the presence and number of orbifold singularities on the boundary of an orbisurface and it detects the number each of smooth and singular boundary components. Moreover, we find that the Steklov spectrum also determines the lengths of the boundary components modulo an equivalence relation, and we show by examples that this result is the best possible. We construct various examples of Steklov isospectral Riemannian orbifolds which demonstrate that these two-dimensional results do not extend to higher dimensions. In addition, we give two- imensional examples which show that the Steklov spectrum does \emph{not} detect the presence of interior singularities nor does it determine the orbifold Euler characteristic. In fact, a flat disk is Steklov isospectral to a cone. In another direction, we obtain upper bounds on the Steklov eigenvalues of a Riemannian orbifold in terms of the isoperimetric ratio and a conformal invariant. We generalize results of B. Colbois, A. El Soufi and A. Girouard, and the fourth author to the orbifold setting; in the process, we gain a sharpness result on these bounds that was not evident in the manifold setting. In dimension two, our eigenvalue bounds are solely in terms of the orbifold Euler characteristic and the number each of smooth and singular boundary components
Dixmier traces on noncompact isospectral deformations
We extend the isospectral deformations of Connes, Landi and Dubois-Violette
to the case of Riemannian spin manifolds carrying a proper action of the
noncompact abelian group . Under deformation by a torus action, a standard
formula relates Dixmier traces of measurable operators to integrals of
functions on the manifold. We show that this relation persists for actions of
, under mild restrictions on the geometry of the manifold which guarantee
the Dixmier traceability of those operators.Comment: 30 pages, no figures; several minor improvements, to appear in J.
Funct. Ana
On Systolic Zeta Functions
We define Dirichlet type series associated with homology length spectra of
Riemannian, or Finsler, manifolds, or polyhedra, and investigate some of their
analytical properties. As a consequence we obtain an inequality analogous to
Gromov's classical intersystolic inequality, but taking the whole homology
length spectrum into account rather than just the systole
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