arXiv research

Proves embedding of metric spaces into Lorentzian space.

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.

Non-convex extremal length found in surface metrics.

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…

Synthetic approach to conformal transformations in metric and Lorentzian spaces.

Teichmüller spaces almost match under specific metrics.

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…

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

Characterizes intrinsic Lorentzian spaces using midpoint properties.

There are several Teichmüller spaces associated to a surface of infinite topological type, after the choice of a particular basepoint (a complex or a hyperbolic structure on the surface). These spaces include the quasiconformal Teichmüller space, the length spectrum Teichmüller space, the Fenchel-Nielsen Teichmüller sp…

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…

Researchers refine local rigidity for marked length spectrum and introduce a new pressure metric.

Random walks on metric spaces embed quasi-isometrically into the space.

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

We consider some metrics and weak metrics defined on the Teichmueller space of a surface of finite type with nonempty boundary, that are defined using the hyperbolic length spectrum of simple closed curves and of properly embedded arcs, and we compare these metrics and weak metrics with the Teichmüller metric. The comp…

Study sub-Riemannian metrics on compact Lie groups, finding same shortest loops.

We define and study metrics and weak metrics on the Teichmueller space of a surface of topologically finite type with boundary. These metrics and weak metrics are associated to the hyperbolic length spectrum of simple closed curves and of properly embedded arcs in the surface. We give a comparison between the defined m…

Geodesic currents on surfaces have comparable metrics in thick regions.

Study on metrics on Teichmüller space of one-holed tori.

We show that the Teichmüller space of a surface without boundary and with punctures, equipped with Thurston's metric is the limit (in an appropriate sense) of Teichmüller spaces of surfaces with boundary, equipped with their arc metrics, when the boundary lengths tend to zero. We use this to obtain a result on the tran…

Bing and Moise proved, independently, that any Peano continuum admits a length metric d. We treat non-degenerate Peano continua with a length metric as evolution systems instead of stationary objects. For any compact length space (X, d) we consider a semiflow in the hyperspace $2^X$ of all non-empty closed sets in X. T…

New method glues Lorentzian spaces, preserving curvature bounds.

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…

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

Let $X_0$ be a complete hyperbolic surface of infinite type that has a geodesic pants decomposition with cuff lengths bounded above. The length spectrum Teichmüller space $T_{ls}(X_0)$ consists of homotopy classes of hyperbolic metrics on $X_0$ such that the ratios of the corresponding simple closed geodesic for the hy…

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…

The $H^1(ds)$-gradient flow shrinks circles with radius $r_0$ to a point.

New method proves length spectrum rigidity in various geometric settings.

Defines new metrics for Lorentzian spaces and their convergence.

Characterizes metrics on triangulated surfaces using glued Euclidean triangles.

Study proves uniqueness of hyperbolic cone structures up to isotopy.

Study geodesic structure of compact balls space and find explicit isometry.

We study side-lengths of triangles in path metric spaces. We prove that unless such a space X is bounded, or quasi-isometric to line or half-line, every triple of real numbers satisfying the strict triangle inequalities, is realized by the side-lengths of a triangle in X. We construct an example of a complete path metr…

Study geodesic discs with boundary length bounds, finding their closure in metric space.

Introduces length space theory in Lorentzian geometry.

Cantor Riemannium is a new type of space from holomorphic germs.

This paper, the second of a series, deals with the function space of all smooth Kähler metrics in any given closed complex manifold $M$ in a fixed cohomology class. The previous result of the second author \cite{chen991} showed that the space is a path length space and it is geodesically convex in the sense that any tw…

Study geodesic extendibility on metric spaces and map them to a half-space.

Study elastic metrics on loops, simplifying reparameterization issues.

New inequalities link boundary lengths to orthogeodesics in symmetric spaces.

Defines magnitude for length spaces with measures, agreeing with finite spaces' magnitude.

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…

New spectra defined for metric spaces, extending existing covering spectrum.

In this paper we consider flat metrics (semi-translation structures) on surfaces of finite type. There are two main results. The first is a complete description of when a set of simple closed curves is spectrally rigid, that is, when the length vector determines a metric among the class of flat metrics. Secondly, we gi…

Study the energy spectrum of metrics on surfaces and its relation to simple length spectrum.

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

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

We study the Teichmüller metric on the Teichmüller space of a surface of finite type, in regions where the injectivity radius of the surface is small. The main result is that in such regions the Teichmüller metric is approximated up to bounded additive distortion by the sup metric on a product of lower dimensional spac…