arXiv research

Study gluing of Lorentzian length spaces and their causal ladder properties.

Study extends null distance concept to Lorentzian length spaces for spacetime analysis.

Given a surface of infinite topological type, there are several Teichmüller spaces associated with it, depending on the basepoint and on the point of view that one uses to compare different complex structures. This paper is about the comparison between the quasiconformal Teichmüller space and the length-spectrum Teichm…

Extremal length is an important conformal invariant on Riemann surface. It is closely related to the geometry of Teichmuller metric on Teichmuller space. By identifying extremal length functions with energy of harmonic maps from Riemann surfaces to $\mathbb{R}$-trees, we study the second variation of extremal length fu…

We prove a strong multiplicity one theorem for the length spectrum of compact even dimensional hyperbolic spaces i.e. if all but finitely many closed geodesics for two compact even dimensional hyperbolic spaces have the same length, then all closed geodesics have the same length.

The study examines how quadratic inequalities affect distances in length spaces.

Establishes a version of Bartnik's conjecture for Lorentzian length spaces.

Timelike curves in Lorentzian length spaces have a total curvature notion that agrees with smooth curves.

Synthetic approach to conformal transformations in metric and Lorentzian spaces.

Study defines new products for Lorentzian spaces and analyzes causal diamonds.

Generalizes Toponogov theorem to Alexandrov spaces.

The paper proves rigidity of length identities for simple closed curves on hyperbolic surfaces.

We prove that the length spectrum metric and the arc-length spectrum metric are almost-isometric on the $ε_0$-relative part of Teichmuller spaces of surfaces with boundary.

Characterizes curves with short representatives on hyperbolic surfaces.

Study causal structure of warped spacetimes using novel pre-length spaces.

Study on Lorentzian spaces with curvature bounds, proving comparison theorems.

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 investigat…

Random walks on hyperbolic spaces show linear growth in translation lengths.

Non-convex extremal length found in surface metrics.

We find the minimal value of the length in de Sitter space of closed space-like curves with non-vanishing non-space-like geodesic curvature vector. These curves are in correspondence with closed almost-regular canal surfaces, and their length is a natural magnitude in conformal geometry. As an application, we get a low…

Given a compact orientable surface of negative Euler characteristic, there exists a natural pairing between the Teichmueuller space of the surface and the set of homotopy classes of simple loops and arcs. The length pairing sends a hyperbolic metric and a homotopy class of a simple loop or arc to the length of geodesic…

Study bounds the length of shortest periodic geodesics on certain curved spaces.

The study constructs a Lorentzian length space and explores its properties and relationships with metric and causal geometry.

As in a symmetric space of noncompact type, one can associate to an oriented geodesic segment in a Euclidean building a vector valued length in the Euclidean Weyl chamber; in addition to the metric length it contains information on the direction of the segment. We study in this paper restrictions on the vector valued s…

The aim of this (mostly expository) article is twofold. We first explore a variety of length functions on the space of currents, and we survey recent work regarding applications of length functions to counting problems. Secondly, we use length functions to provide a proof of a folklore theorem which states that pseudo-…

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 lattice extensions of Schottky groups in hyperbolic space.

New results on the convexity of geodesic-length functions on Teichmüller space are presented. A formula for the Hessian of geodesic-length is presented. New bounds for the gradient and Hessian of geodesic-length are described. A relationship of geodesic-length functions to Weil-Petersson distance is described. Applicat…

New method glues Lorentzian spaces, preserving curvature bounds.

Globalisation theorem for Lorentzian spaces with curvature bounds.

Study finds minimum lengths of curves on a one-holed torus.

We prove an analogue of Farb-Masur's theorem that the length-spectra metric on moduli space is "almost isometric" to a simple model $\mathcal {V}(S)$ which is induced by the cone metric over the complex of curves. As an application, we know that the Teichmüller metric and the length-spectra metric are "almost isometric…

Constructs non-isometric iso-length-spectral surfaces.

New method proves length spectrum rigidity in various geometric settings.

Curvature bounds preserved in length-minimizing disks.

Killing vector fields of constant length correspond to isometries of constant displacement. Those in turn have been used to study homogeneity of Riemannian and Finsler quotient manifolds. Almost all of that work has been done for group manifolds or, more generally, for symmetric spaces. This paper extends the scope of …

In Euclidean geometry, all metric notions (arc length for curves, the first fundamental form for surfaces, etc.) are derived from the Euclidean inner product on tangent vectors, and this inner product is preserved by the full symmetry group of Euclidean space (translations, rotations, and reflections). In equiaffine ge…

Spaces with similar long paths have similar shapes.

Characterizes paths minimizing anisotropic lengths in Euclidean space.

The paper examines conditions for compactness in sequences of warped product length spaces.

The abstract discusses a new type of space and its properties.

Synthetic proof shows globally hyperbolic Lorentzian spaces with specific curvature are warped products.

We prove that there are Fenchel-Nielsen coordinates for the Teichmueller space of a finite area hyperbolic surface with respect to which the length functions are convex.

It is well-known that the class of piecewise smooth curves together with a smooth Riemannian metric induces a metric space structure on a manifold. However, little is known about the minimal regularity needed to analyze curves and particularly to study length-minimizing curves where neither classical techniques such as…

Characterizes intrinsic Lorentzian spaces using midpoint properties.

We define a Hamilton-Jacobi semigroup acting on continuous functions on a compact length space. Following a strategy of Bobkov, Gentil and Ledoux, we use some basic properties of the semigroup to study geometric inequalities related to concentration of measure. Our main results are that (1) a Talagrand inequality on a …

A general approach to proving that the length spectrum of a compact Riemannian manifold is an invariant of the Laplace spectrum comes from considering the wave trace, a spectrally determined tempered distribution. The Poisson relation states that the singularities of the wave trace can only occur at lengths of closed g…

The paper extends Reshetnyak's theorem to Lorentzian length spaces with upper curvature bounds.