arXiv research

New theorem shows metrics of certain groups are close if their lengths are identical.

Paper shows non-arithmetic surface with unique geometric property.

Study approximate marked length spectrum rigidity in non-positively curved groups.

Study proves rigidity of marked length spectra in contracting group actions.

New theorem shows certain curved surfaces are uniquely identified by their geodesic lengths.

Anosov surfaces with same length spectrum are isometric.

In this paper we prove that the space of flat metrics (nonpositively curved Euclidean cone metrics) on a closed, oriented surface is marked length spectrally rigid. In other words, two flat metrics assigning the same lengths to all closed curves differ by an isometry isotopic to the identity. The novel proof suggests a…

Study shows surfaces with similar length spectra are smoothly deformable.

New method proves length spectrum rigidity in various geometric settings.

Two Anosov metrics with same boundary distance are isometric.

The study finds sparse sets that uniquely determine metrics on negatively curved manifolds.

Smoothly conjugate Anosov flows on 3D manifolds are actually smoothly conjugate.

Combining several previously known arguments, we prove marked length spectrum rigidity for surfaces with nonpositively curved Riemannian metrics away from a finite set of cone-type singularities with cone angles $>2π$. With an additional condition, we can weaken the requirement on one metric to `no conjugate points.'

The paper proves uniqueness of closed timelike geodesics on de-Sitter tori with one singularity.

The paper explores how magnetic systems' spectra can identify metrics and 1-forms.

In all dimensions, we prove that the marked length spectrum of a Riemannian manifold $(M,g)$ with Anosov geodesic flow and non-positive curvature locally determines the metric in the sense that two close enough metrics with the same marked length spectrum are isometric. In addition, we provide a completely new stabilit…

We prove that for compact, non-contractible, one dimensional geodesic spaces, a version of the marked length spectrum conjecture holds. For a compact one dimensional geodesic space X, we define a subspace Conv(X). When X is non-contractible, we show that X deformation retracts to Conv(X). If two such spaces X, Y have t…

New stability estimate for metric rigidity in hyperbolic dynamics.

Closed manifolds with close marked spectra are approximately isometric.

The rigidity of marked length spectrum for closed hyperbolic surfaces due to Fricke-Klein [7] has been the motivation of many different rigidity results, specially for manifolds of negative curvature. From the works of Vigneras [18], Sunada [17] and many other authors this result is far from being true for the unmarked…

New rigidity result for convex co-compact actions in products of spaces.

We refine the recent local rigidity result for the marked length spectrum obtained by the first and third author in \cite{Guillarmou-Lefeuvre-18} and give an alternative proof using the geodesic stretch between two Anosov flows and some uniform estimate on the variance appearing in the central limit theorem for Anosov …

Any two compact, complete, one-dimensional geodesic spaces with identical marked length spectrum have isometric $π_1$-hull. The present version contains errors, notably in Lemmas 2.2 and 2.3 (path cancellations can be more complicated), which then propagate through the paper. The main result is correct as stated, and a…

In this paper we show that if the limit set is not small ,marked length spectrum determines geometric structure of rank one locally symmetric manifolds.

Given a closed hyperbolic Riemannian surface, the aim of the present paper is to describe an explicit construction of smooth deformations of the hyperbolic metric into Finsler metrics that are not Riemannian and whose properties are such that the classical Riemannian results about entropy rigidity, marked length spectr…

This paper studies spectral properties of spheres with one equator.

New counterexample shows curved surfaces can deform geodesics without diffeomorphism.

Study of group actions on CAT(0) cube complexes, focusing on marked length spectra.

Study magnetic geodesics on Heisenberg groups and manifolds.

New abelian cohomology theory applied to rigidity of flows.

We prove that a Kleinian surface groups is determined, up to conjugacy in the isometry group of $\mathbb H^3$, by its simple marked length spectrum. As a first application, we show that a discrete faithful representation of the fundamental group of a compact, acylindrical, hyperbolizable 3-manifold $M$ is similarly det…

The paper generalizes rigidity results for contact Anosov flows with bunching assumption.

We show that group actions on irreducible ${\rm CAT(0)}$ cube complexes with no free faces are uniquely determined by their $\ell^1$ length function. Actions are allowed to be non-proper and non-cocompact, as long as they are minimal and have no finite orbit in the visual boundary. This is, to our knowledge, the first …

The paper proves a rigidity theorem for non-compact convex sets in hyperbolic 3-space.

New magnetic flow rigidity theorem for negative curvatures.

We study the geometry of hyperbolic cone surfaces, possibly with cusps or geodesic boundaries. We prove that any hyperbolic cone structure on a surface of non-exceptional type is determined up to isotopy by the geodesic lengths of a finite specific homotopy classes of non-peripheral simple closed curves. As an applicat…

The subject of this paper is the relationship among the marked length spectrum, the length spectrum, the Laplace spectrum on functions, and the Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show that for a large class of three-step nilmanifolds, if a pair of nilmanifolds in this class has the …

The subject of this paper is the relationship among the marked length spectrum, the length spectrum, the Laplace spectrum on functions, and the Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show that for a large class of three-step nilmanifolds, if a pair of nilmanifolds in this class has the …

Metrics are isometric for certain Anosov magnetic systems.

We describe a method of encoding various types of link diagrams, including those with classical, flat, rigid, welded, and virtual crossings. We show that this method may be used to encode link diagrams, up to equivalence, in a notation whose length is a cubic function of the number of 'riser marks'. For classical knots…

Under the assumption that the X-ray transform over symmetric solenoidal 2-tensors is injective, we prove that smooth compact connected manifolds with strictly convex boundary, no conjugate points and a hyperbolic trapped set are locally marked boundary rigid.

This article is the second in a series of two whose aim is to extend a recent result of Guillarmou-Lefeuvre [arXiv:1806.04218] on the local rigidity of the marked length spectrum from the case of compact negatively-curved Riemannian manifolds to the case of manifolds with hyperbolic cusps. We deal with the nonlinear ve…

New method uses short geodesics to approximate marked length spectrum.

Spaces with similar long paths have similar shapes.

We show, using standard results in length spectrum rigidity and symplectic homology, that if the unit tangent bundles of two compact surfaces of negative curvature are exact symplectomorphic, then the underlying surfaces are isometric, and hence the disk bundles are symplectomorphic smoothly up to the boundaries. Motiv…

Study geometric rigidity of surfaces in negative curvature manifolds.

This paper is the first in a series of two articles whose aim is to extend a recent result of Guillarmou-Lefeuvre on the local rigidity of the marked length spectrum from the case of compact negatively-curved Riemannian manifolds to the case of manifolds with hyperbolic cusps. In this first paper, we deal with the line…

Algorithms compute length spectra of torus graphs efficiently.