arXiv research

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…

Extends curve functions to geodesic currents with a simple criterion.

Study geodesic paths on flat surfaces, comparing length and singularity counts.

Research extends geodesic length function study to three holed sphere.

The paper proves geodesics and conic sections are length-minimizing under specific metrics.

Study geodesics on flat tori, focusing on orthogonal lengths and their distribution.

New approach finds minima of geodesic lengths for non-uniform fillings.

Upper bound established for the length of shortest closed geodesics in hyperbolic link complements.

Using geodesic length functions, we define a natural family of real codimension 1 subvarieties of Teichmüller space, namely the subsets where the lengths of two distinct simple closed geodesics are of equal length. We investigate the point set topology of the union of all such hypersurfaces using elementary methods. Fi…

Analyzes geodesic lengths in sparse networks, deriving a distribution.

We show that for every simple closed curve α, the extremal length and the hyperbolic length of αare quasi-convex functions along any Teichmuller geodesic. As a corollary, we conclude that, in Teichmuller space equipped with the Teichmuller metric, balls are quasi- convex.

Study minima of geodesic lengths for specific curves on surfaces.

Proves existence of curves with constant curvature in a sphere.

Geodesic currents on surfaces have comparable metrics in thick regions.

Study shows shortest geodesic length on certain manifolds is limited by volume, diameter, and cover elements.

We investigate the distribution of lengths obtained by intersecting a random geodesic with a geodesic lamination. We give an explicit formula for the distribution for the case of a maximal lamination and show that the distribution is independent of the surface and lamination. We also show how the moments of the distrib…

We give an identity involving sums of functions of lengths of simple closed geodesics, known as a McShane identity, on any non-orientable hyperbolic surface with boundary which generalises Mirzakhani's identities on orientable hyperbolic surfaces with boundary.

Given a compact orientable surface with finitely many punctures $Σ$, let $\Cal S(Σ)$ be the set of isotopy classes of essential unoriented simple closed curves in $Σ$. We determine a complete set of relations for a function from $\Cal S(Σ)$ to $\bold R$ to be the geodesic length function of a hyperbolic metric with geo…

In a family of compact, canonically polarized, complex manifolds equipped with Kähler-Einstein metrics the first variation of the lengths of closed geodesics was previously shown in by the authors in [arXiv:0808.3741v2] to be the geodesic integral of the harmonic Kodaira-Spencer form. We compute the second variation. F…

Let $\mathcal T$ be the Teichmüller space of marked genus $g$, $n$ punctured Riemann surfaces with its bordification $\Tbar$ the {\em augmented Teichmüller space} of marked Riemann surfaces with nodes, \cite{Abdegn, Bersdeg}. Provided with the WP metric $\Tbar$ is a complete CAT(0) metric space, \cite{DW2, Wlcomp, Yam2…

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…

The paper studies gradients of geodesic-length functions and systoles on Teichmüller spaces.

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

We find a canonical decomposition of a geodesic current on a surface of finite type arising from a topological decomposition of the surface along special geodesics. We show that each component either is associated to a measured lamination or has positive systole. For a current with positive systole, we show that the in…

Let $S$ be a closed oriented surface of genus at least $2$, and denote by $\mathcal{T}(S)$ its Teichm{ü}ller space. For any isotopy class of closed curves $γ$, we compute the first three derivatives of the length function $\ell\_γ:\mathcal{T}(S)\rightarrow\mathbf{R}\_+$ in the shearing coordinates associated to a maxim…

Characterizes paths minimizing anisotropic lengths in Euclidean space.

Formula for integrating random variables on hyperbolic surfaces.

New method uses short geodesics to approximate marked length spectrum.

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-…

New characterization of geodesic currents via curve functionals.

Sharp bounds on mean curvature and geodesic lengths in convex hypersurfaces.

This paper discovers new identities linking geodesic and orthogeodesic lengths on hyperbolic surfaces.

In this paper we establish a relationship between geodesic nets and critical points of the distance function. We bound the number of balanced points for certain minimizing geodesic nets on manifolds homeomorphic to the $n$-sphere. We also bound the length of certain minimizing geodesic nets.

Study geodesics on flat tori, focusing on convex bodies.

Study finds geodesic networks for surfaces with convex boundary.

In this paper we study 1/k geodesics, those closed geodesics that minimize on all subintervals of length $L/k$, where $L$ is the length of the geodesic. We develop new techniques to study the minimizing properties of these curves on doubled polygons, and demonstrate a sequence of doubled polygons whose closed geodesics…

Study counts geodesics on modular surface, linking to necklace counting.

We show that the length function of a measured geodesic lamination is convex in Thurston's shear coordinates over Teichmüller space and strictly convex for generic laminations. We give some consequences of this result in the context of Thurston's asymmetric metric on Teichmüller space.

Wolpert's cosine formula on Teichmüller space gives the Weil-Petersson Poisson bracket $\{l_α, l_β\}$ for geodesic length functions $l_α,l_β$ of closed curves $α,β$ as the sum of the cosines of the angle of intersection of the associated geodesics. This was recently generalized to Hitchin representations by Labourie. I…

Study on lengths and curvatures of harmonic functions on smooth and singular surfaces.

Improved bounds on geodesic lengths in Riemannian surfaces.

We prove that a certain series defines a constant function using Wolpert's formula for the variation of the length of a geodesic along a Fenchel Nielsen twist. Subsequently we determine the value viewing it as function on the the Deligne Mumford compactification and evaluating it at the stable curve at infinity.

The paper establishes bounds on the lengths of geodesics on manifolds with curvature constraints.

Luo and Tan gave a new identity for hyperbolic surfaces with/without geodesic boundary in terms of dilogarithms of the lengths of simple closed geodesics on embedded three-holed spheres or one-holed tori. However, the identity was trivial for a hyperbolic one-holed torus with geodesic boundary. In this paper we adapt t…

Lower bounds on geodesic length with few intersections on hyperbolic surfaces.

Proves stability of convex spheres with similar geodesic lengths.

Geodesic concavity and hypersymplectic structures in $G2$-structures space.

New asymmetric metric on Teichmüller space for surfaces.