arXiv research

Filling length measures the length of the contracting closed loops in a null-homotopy. The filling length function of Gromov for a finitely presented group measures the filling length as a function of length of edge-loops in the Cayley 2-complex. We give a bound on the filling length function in terms of the log of an …

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…

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…

Link between Teichmüller and anti de Sitter geometry via length functions.

Introduces a new length functional for Ricci flow to detect steady solitons.

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-…

Extends curve functions to geodesic currents with a simple criterion.

In this paper, we show that the extremal length functions on Teichmüller space are log-plurisubharmonic. As a corollary, we obtain an alternative proof of L.Liu and W.Su's results on the plurisubharmonicity of extremal length functions. We also obtain alternative proofs of S.Krushkal's results that a function defined b…

Study length functions on various groups and prove homomorphisms to finite groups.

Relative cup-length defined for non-Morse functions on manifolds.

We study metric and analytic properties of generalized lemniscates E_t(f)={z:ln|f(z)|=t}, where f is an analytic function. Our main result states that the length function |E_t(f)| is a bilateral Laplace transform of a certain positive measure. In particular, the function ln|E_t(f)| is convex on any interval free of cri…

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

Recurrent neural networks and sequence to sequence models require a predetermined length for prediction output length. Our model addresses this by allowing the network to predict a variable length output in inference. A new loss function with a tailored gradient computation is developed that trades off prediction accur…

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

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

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 …

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

Research extends geodesic length function study to three holed sphere.

For a surface group, a new bound is given for conjugator length function.

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.

New rigidity result for hyperbolic surfaces based on curve lengths.

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…

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

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

In this paper we raise the question whether every closed Riemannian manifold has a spine of minimal area, and we answer it affirmatively in the surface case. On constant curvature surfaces we introduce the spine systole, a continuous real function on moduli space that measures the minimal length of a spine in each surf…

The paper studies stability of discrete planar curves using variational methods.

Following \cite{citeSavelyevVirtualMorsetheoryon$Omega$Ham$(Momega)$.}, we develop here a connection between Morse theory for the (positive) Hofer length functional $L: Ω\text {Ham}(M, ω) \to \mathbb{R}$, with Gromov-Witten/Floer theory, for monotone symplectic manifolds $(M, ω)$. This gives some immediate restrictio…

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

We calculate the higher derivatives of length functions on Teichmuller space along earthquake deformations. This generalizes the cosine formula for the first derivative by Kerckhoff and Wolpert and the sine formula for second derivative by Wolpert.

A pseudo-length function defined on an arbitrary group $G = (G,\cdot,e, (\,)^{-1})$ is a map $\ell: G \to [0,+\infty)$ obeying $\ell(e)=0$, the symmetry property $\ell(x^{-1}) = \ell(x)$, and the triangle inequality $\ell(xy) \leqslant \ell(x) + \ell(y)$ for all $x,y \in G$. We consider pseudo-length functions which sa…

We study the spectral properties of a large class of compact flat Riemannian manifolds of dimension 4, namely, those whose corresponding Bieberbach groups have the canonical lattice as translation lattice. By using the explicit expression of the heat trace of the Laplacian acting on $p$-forms, we determine all $p$-isos…

The systole length of hyperbolic n-manifolds is bounded by a function of n and t.

Length metrics can be closely approximated by conformally flat metrics.

Non-convex extremal length found in surface metrics.

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…

Study shows surfaces with similar length spectra are smoothly deformable.

We introduce here a natural functional associated to any $b \in QH_* (M, ω)$: \emph{spectral length functional}, on the space of "generalized paths" in $\text {Ham}(M, ω)$, closely related to both the Hofer length functional and spectral invariants and establish some of its properties. This functional is smooth on its…

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.

Given a Hitchin representation $ρ\colon π_1(S) \to \PSL_n(\mathbb{R})$, we construct $n$ continuous functions $\ell_i^ρ\colon \mathcal \CH(S) \to \mathbb{R}$ defined on the space of Hölder geodesic currents $\CH(S)$ such that, for a closed, oriented curve $γ$ in $S$, the $i$--th eigenvalue of the matrix $ρ(γ)\in \PSL_n…

Formula connects length and correlation functions via ghost polygons and Poisson bracket.

Critical trajectories in a sphere are found for a specific bending functional.

In this paper, we determine geometric information on slope lengths of a large class of knots in the 3-sphere, based only on diagrammatical properties of the knots. In particular, we show such knots have meridian length strictly less than 4, and we find infinitely many families with meridian length approaching 4 from be…

ReLU networks don't exponentially distort curve lengths as previously thought.

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.

New algorithm reduces regret bounds for Bayesian optimization with unknown hyperparameters.

Much recent work has concerned sparse approximations to speed up the Gaussian process regression from the unfavorable O(n3) scaling in computational time to O(nm2). Thus far, work has concentrated on models with one covariance function. However, in many practical situations additive models with multiple covariance func…

The paper analyzes discrete approximations to minimize curve length in Euclidean space.