Marked Length Spectrum Research Articles

The marked length spectrum of a projective manifold or orbifold

A strictly convex real projective orbifold is equipped with a natural Finsler metric called a Hilbert metric. In the case that the projective structure is hyperbolic, the Hilbert metric and the hyperbolic metric coincide. We prove that the marked Hilbert length spectrum determines the projective structure only up to projective duality. A corollary is the existence of non-isometric diffeomorphic strictly convex projective manifolds (and orbifolds) that are isospectral. This corollary follows from work of Goldman and Choi, and Benoist.

Some smooth Finsler deformations of hyperbolic surfaces

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 spectrum rigidity and boundary rigidity all fail to extend to the Finsler category.

Symplectic rigidity for Anosov hypersurfaces

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. Motivated by the work of Eliashberg and Hofer on unseen symplectic boundaries, we try to extend the above result to other classes of domains in (co)tangent bundles of surfaces. The domains we can treat consist of those bounded by deformations of negatively curved unit cotangent bundles through boundaries with Anosov characteristic (Reeb) flows. We show that if two such are exact symplectomorphic, then the two domains are symplectomorphic by a diffeomorphism smooth up to the boundary. Furthermore, if two such surfaces have the same marked length spectrum then they are connected by a 1-parameter family of domains bounded by hypersurfaces with smoothly time preserving conjugate Reeb flows by a smooth family of Hamiltonian diffeomorphisms. A few counterexamples to easy generalizations are given.

Free Energy as a Geometric Invariant

The free energy plays a fundamental role in statistical and condensed matter physics. A related notion of free energy plays an important role in the study of hyperbolic dynamical systems. In this paper we introduce and study a natural notion of free energy for surfaces with variable negative curvature. This geometric free energy encodes a new type of marked length spectrum of closed geodesics, which lies between the well-known marked length spectrum (marked by the corresponding element of the fundamental group) and the unmarked length spectrum. We prove that the free energy parametrizes the boundary of the domain of convergence of a Poincare series which also encodes this spectrum. We also show that this new length spectrum, or equivalently the geometric free energy, is not an isometry invariant. In the final section we use tools from multifractal analysis to effect a fine asymptotic comparison of word length and geodesic length of closed geodesics. We hope that our geometric understanding of free energy will provide new insight into this fundamental physical and dynamical quantity.

The minimal marked length spectrum of Riemannian two-step nilmanifolds

The purpose of this paper is to compare the minimal marked length spectrum and the Laplace spectrum on functions and on forms for Riemannian two-step nilmanifolds. A Riemannian nilmanifold is a closed manifold of the form ( \G, g), where G is a simply connected nilpotent Lie group, is a cocompact (i.e., \G compact) discrete subgroup of G, and g arises from a left invariant metric on G. Examples of nilmanifolds include flat tori and Heisenberg manifolds. The Laplace spectrum of a closed Riemannian manifold (M, g) is the set of eigenvalues of the Laplace–Beltrami operator , counted with multiplicity. The Laplace–Beltrami operator may be extended to act on smooth p-forms by = dδ + δd, where δ is the metric adjoint of d. Two manifolds have the same marked length spectrum if there exists an isomorphism between the fundamental groups such that corresponding free homotopy classes of loops can be represented by smoothly closed geodesics of the same length. Two manifolds have the same minimal marked length spectrum (resp., maximal marked length spectrum) if there exists an isomorphism between the fundamental groups such that the smallest (resp., longest) closed loops in corresponding free homotopy classes have the same length. The main theorem of this paper is the following (see Theorem 2.5).

The covering spectrum of a compact length space

We define a new spectrum for compact length spaces and Riemannian manifolds called the “covering spectrum” which roughly measures the size of the one dimensional holes in the space. More specifically, the covering spectrum is a set of real numbers δ>0 which identify the distinct δ covers of the space. We investigate the relationship between this covering spectrum, the length spectrum, the marked length spectrum and the Laplace spectrum. We analyze the behavior of the covering spectrum under Gromov–Hausdorff convergence and study its gap phenomenon.

Rigidity on symmetric spaces

In this note we show the marked length rigidity of symmetric spaces. More precisely, if X and Y are symmetric spaces of noncompact type without Euclidean de Rham factor, with G 1 and G 2 corresponding real semisimple Lie groups, and Γ 1⊂ G 1, Γ 2⊂ G 2 are Zariski dense subgroups with the same marked length spectrum, then X= Y and Γ 1, Γ 2 are conjugate by an isometry. As an application, we answer in the affirmative a Margulis's question and show that the cross-ratio on the limit set determines the Zariski dense subgroups up to conjugacy. We also embed the space of nonparabolic representations from Γ to G into R Γ .

Marked length rigidity of rank one symmetric spaces and their product

In this paper we show that if two Zariski dense representations, from a group G into Iso( X) where X is rank one symmetric space, have the proportional marked length spectrum, then they are conjugate. As a generalization we show that a Zariski dense representation into the isometry group of the product of rank one symmetric spaces is determined by the marked cross ratio.

Rigidity and Deformation Spaces of Strictly Convex Real Projective Structures on Compact Manifolds

In this paper we show that if two strictly convex, compact real projective manifolds have the same marked length spectrum with respect to the Hilbert metric, then they are projectively equivalent. This is a rigidity for Finsler metric with a special geometric structure. Furthermore we prove an analogue of a Hitchin's conjecture for hyperbolic 3-manifolds, namely the deformation space of convex real projective structures on a compact hyperbolic 3-manifold M is a component in the moduli space of PGL(4,ℍ)-representations of π1(M).

Ergodic theory and rigidity on the symmetric space of non-compact type

In this paper we investigate the rigidity of symmetric spaces of non-compact type using ergodic theory such as Patterson–Sullivan measure and the marked length spectrum along with the cross ratio on the limit set. In particular, we prove that the marked length spectrum determines the Zariski dense subgroup up to conjugacy in the isometry group of the product of rank-one symmetric spaces. As an application, we show that two convex cocompact, negatively curved, locally symmetric manifolds are isometric if the Thurston distance is zero and the critical exponents of the Poincaré series are the same, and the same is true if the geodesic stretch is equal to one.

Continuous families of Riemannian manifolds, Isospectral on functions but not on 1-forms

The purpose of this paper is to present the first continuous families of Riemannian manifolds that are isospectral on functions but not on 1-forms, and, simultaneously, the first continuous families of Riemannian manifolds with the same marked length spectrum but not the same 1-form spectrum. Examples of isospectral manifolds that are not isospectral on forms are sparse, as most examples of isospectral manifolds can be explained by Sunada’s method or its generalizations, hence are strongly isospectral. The examples here are three-step Riemannian nilmanifolds, arising from a general method for constructing isospectral Riemannian nilmanifolds previously presented by the author. Gordon and Wilson constructed the first examples of nontrivial isospectral deformations, continuous families of Riemannian nilmanifolds. Isospectral manifolds constructed using the Gordon-Wilson method, a generalized Sunada method, are strongly isospectral and must have the same marked length spectrum. Conversely, Ouyang and Pesce independently showed that all isospectral deformations of two-step nilmanifolds must arise from the Gordon-Wilson method, and Eberlein showed that all pairs of two-step nilmanifolds with the same marked length spectrum must come from the Gordon-Wilson method.

Delocalized L2-Invariants

We define extensions of the L2-analytic invariants of closed manifolds, called delocalized L2-invariants. These delocalized invariants are constructed in terms of a nontrivial conjugacy class of the fundamental group. We show that in many cases, they are topological in nature. We show that the marked length spectrum of an odd-dimensional hyperbolic manifold can be recovered from its delocalized L2-analytic torsion. There are technical convergence questions.

Coordinates of the representation space in the semisimple Lie group of rank one

In this paper we show that the space of irreducible representations from a finitely presented group into the group of isometries of a rank one symmetric space of non-compact type, embeds into ℝn for some n, where the coordinates are the translation lengths of isometries in the representation. The ingredients of the proof consist of the two facts that the representation is determined by its marked length spectrum and that the nested sequence of algebraic subvarieties is stabilised at a finite step by the Noetherian property of the polynomial ring. As a minor application, we use this fact to simplify McMullen's proof about the exponential algebraic convergence of Thurston's double limit to the geometrically infinite manifold in the space of discrete faithful representations of π1(S) in Iso+.

The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles

We compute the spectrum of the Dirac operator on 3-dimensional Heisenberg manifolds. The behavior under collapse to the 2-torus is studied. Depending on the spin structure either all eigenvalues tend to1 or there are eigenvalues converging to those of the torus. This is shown to be true in general for collapsing circle bundles with totally geodesic bers. Using the Hopf bration we use this fact to compute the Dirac eigenvalues on complex projective space including the multiplicities. Finally, we show that there are 1-parameter families of Riemannian nilmanifolds such that the Laplacian on functions and the Dirac operator for certain spin structures have constant spectrum while the Laplacian on 1-forms and the Dirac operator for the other spin structures have nonconstant spectrum. The marked length spectrum is also constant for these families.

On the rigidity of discrete isometry groups of negatively curved spaces

We prove an ergodic rigidity theorem for discrete isometry groups of CAT(-1) spaces. We give explicit examples of divergence isometry groups with infinite covolume in the case of trees, piecewise hyperbolic 2-polyhedra, hyperbolic Bruhat-Tits buildings and rank one symmetric spaces. We prove that two negatively curved Riemannian metrics, with conical singularities of angles at least $ 2\pi $ , on a closed surface, with boundary map absolutely continuous with respect to the Patterson-Sullivan measures, are isometric. For that, we generalize J.-P. Otal's result to prove that a negatively curved Riemannian metric, with conical singularities of angles at least $ 2\pi $ , on a closed surface, is determined, up to isometry, by its marked length spectrum.

The marked length spectrum vs. the Laplace spectrum on forms on Riemannian nilmanifolds

The subject of this paper is the relationships 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 same marked length spectrum, they necessarily share the same Laplace spectrum on functions. In contrast, we present the first example of a pair of isospectral Riemannian manifolds with the same marked length spectrum but not the same spectrum on one-forms. Outside of the standard spheres vs. the Zoll spheres, which are not even isospectral, this is the only example of a pair of Riemannian manifolds with the same marked length spectrum, but not the same spectrum on forms. This partially extends and partially contrasts the work of Eberlein, who showed that on two-step nilmanifolds, the same marked length spectrum implies the same Laplace spectrum both on functions and on forms.

Rigidity for non-compact surfaces of finite area and certain Kähler manifolds

AbstractWe first consider the rigidity of the marked length spectrum for non-compact surfaces of finite area.

Geodesic rays, Busemann functions and monotone twist maps

We study the action-minimizing half-orbits of an area-preserving monotone twist map of an annulus. We show that these so-called rays are always asymptotic to action-minimizing orbits. In the spirit of Aubry-Mather theory which analyses the set of action-minimizing orbits we investigate existence and properties of rays. By analogy with the geometry of the geodesics on a Riemannian 2-torus we define a Busemann function for every ray. We use this concept to prove that the minimal average action A(α) is differentiable at irrational rotation numbersα while it is generically non-differentiable at rational rotation numbers (cf. also [18]). As an application of our results in the geometric framework we prove that a Riemannian 2-torus which has the same marked length spectrum as a flat 2-torus is actually isometric to this flat torus.

Surfaces with the same marked length spectrum

Various authors have shown that isotopy classes of nonpositively curved Riemannian metrics on surfaces are characterized by their marked length spectrum. We show by an example that, although this property makes sense in a non-Riemannian setting, it does not hold for every metric space structure on a surface. More precisely, given two arbitrary negatively curved Riemannian metrics on a compact surface, we construct a metric which has the same geodesics as the first one, has the same marked length spectrum as the second one, and is in general not isometric to either one.

The marked length-spectrum of a surface of nonpositive curvature

We generalize work of J.P. Otal and C. Croke on the marked length spectrum of surfaces to the case where one of the metrics is of nonpositive curvature and the other one has no conjugate points. If M is a manifold and g1, g2 are two Riemannian metrics, we say that they have the same marked length spectrum if in each homotopy class of closed curves in M the infimum of g1-lengths of curves and the infimum of g2-lengths of curves are the same. The marked length spectrum problem in general is to show that two metrics with the same marked length spectrum are isometric. Of course, this cannot hold for arbitrary metrics (for example if M is simply connected). This problem was stated as a conjecture in [BK] in the case where M is a closed surface and g1 and g2 are of negative curvature. This conjecture was solved by J.P. Otal [To] and independently by C. Croke [Cr]–see also [Fa]. Previous work on the problem was done by Guillemin and Kazhdan [GK]. In this work, using Otal’s approach, we improve some of these results by proving the following theorem: Theorem A. Let M be a closed surface and let g1, g2 be Riemannian metrics on M , with g1 of nonpositive curvature and g2 without conjugate points. If g1 and g2 have the same marked length-spectrum then they are isometric by an isometry homotopic to the identity. We will also prove the following fact, which reduces the length spectrum and curvature condition to the assumption that the Morse correspondence preserves angles—see §1 for the definition of the Morse correspondence. Theorem B. Let M be a closed surface of genus ≥ 2, and let g1, g2 be Riemannian metrics without conjugate points on M . If g1 and g2 have the same marked lengthspectrum and the Morse correspondence preserves angles then they are isometric by an isometry homotopic to the identity.