arXiv research

Describes envelopes of Thurston metric on Teichmüller space.

The Teichmüller space $\mathcal{T}(Σ)$ of a surface $Σ$ is equipped with Thurston's asymmetric metric. Stretch lines are oriented geodesics for this metric on $\mathcal{T}(Σ)$. We give the asymptotic behavior of the lengths of the measured geodesic laminations as one follows a stretch line in the positive direction.

A cylindrical stretch line is a stretch line, in the sense of Thurston, whose horocyclic lamination is a weighted multicurve. In this paper, we show that two correctly parameterized cylindrical lines are parallel if and only if these lines converge towards the same point in Thurston's boundary of Teichmüller space.

We consider the question of whether solutions of variants of Teichmüller harmonic map flow from surfaces $M$ to general targets can degenerate in finite time. For the original flow from closed surfaces of genus at least $2$, as well as the flow from cylinders, we prove that such a finite-time degeneration must occur in…

In 1986 William P. Thurston introduced the celebrated (asymmetric) Lipschitz distance on the Teichmueller space of a (closed or punctured) surface. In this paper we extend his work to the Teichmueller space of a surface with boundary endowed the arc distance. In this new setting we construct a large family of geodesics…

The period map for 4-manifolds is dense and surjective under certain conditions.

Treebolic space HT(q,p) is a key example of a strip complex in the sense of Bendikov, Saloff-Coste, Salvatori, and Woess [Adv. Math. 226 (2011), 992-1055]. It is an analog of the Sol geometry, namely, it is a horocylic product of the hyperbolic upper half plane with a "stretching" parameter q and the homogeneous tree T…

Unique geodesics selected by energy minimization in Teichmüller space.

In the Teichmüller space of a hyperbolic surface of finite type, we construct geodesic lines for Thurston's asymmetric metric having the property that when they are traversed in the reverse direction, they are also geodesic lines (up to reparametrization). The lines we construct are special stretch lines in the sense o…

New examples of harmonic unit vector fields on hyperbolic 3-space are constructed by exploiting the reduction of symmetry arising from the foliation by horospheres. This is compared and contrasted with the analogous construction in Euclidean 3-space, using a foliation by planes, which produces some new examples of harm…

This paper investigates analytic properties of maps between hyperbolic surfaces, focusing on best Lipschitz maps and geodesic laminations.

Abstract notes on hyperbolic surfaces and Teichmüller spaces.

We analyse the asymptotic behaviour of solutions of the Teichmüller harmonic map flow from cylinders, and more generally of `almost minimal cylinders', in situations where the maps satisfy a Plateau-boundary condition for which the three-point condition degenerates. We prove that such a degenerating boundary condition …

Harmonic maps from $\BR^2$ or one-connected domain $Ø\subset \BR^2$ into $GL(m, \BC)$ and $U(m)$ are treated. The GBDT version of the Bäcklund-Darboux transformation is applied to the case of the harmonic maps. A new general formula on the GBDT transformations of the Sym-Tafel immersions is derived. A class of the harm…

There are many industrial situations where rods are used to stir a fluid, or where rods repeatedly stretch a material such as bread dough or taffy. The goal in these applications is to stretch either material lines (in a fluid) or the material itself (for dough or taffy) as rapidly as possible. The growth rate of mater…

Consider oriented surfaces immersed in $\mathbb R^3.$ Associated to them, here are studied pairs of transversal foliations with singularities, defined on the Elliptic region, where the Gaussian curvature $\mathcal K$, given by the product of the principal curvatures $k_1, k_2$ is positive. The leaves of the foliations …

In this note we shall show that the sectional curvature of a harmonic manifold is bounded on both sides. In fact we shall give a pinching constant for all harmonic manifolds. We shall use the imbedding theorem for harmonic manifolds proved by Z.I.Szabo and the description of screw lines in hilbert spaces to prove the r…

New stretch maps minimize distortion in geometric group theory.

Finsler metrics with relatively non-negative (non-positive, respectively), constant and isotropic stretch curvatures are investigated in this paper. In particular, it is proved that every non-Riemannian $(α, β)$-metric with a nonzero constant flag curvature and a non-zero relatively isotropic stretch curvature over a m…

The purpose of this paper is to establish a Nadel vanishing theorem for big line bundles with multiplier ideal sheaves of singular metrics admitting an analytic Zariski decomposition (such as, metrics with minimal singularities and Siu's metrics). For this purpose, we apply the theory of harmonic integrals and generali…

Every weak Perron number is realized as a stretch factor of a homeomorphism on a surface.

Lower bound on stretch factor for periodic maps.

BacHMMachine harmonizes Baroque chorales using theory-driven principles and Hidden Markov Models.

In 1974, Thurston proved that, up to isotopy, every automorphism of closed orientable surface is either periodic, reducible, or pseudo-Anosov. The latter case has lead to a rich theory with applications ranging from dynamical systems to low dimensional topology. Associated with every pseudo-Anosov map is a real number …

The paper constructs non-convergent solutions to Vafa-Witten equations with specific harmonic 2-form limits.

Study stretch laminations in hyperbolic 3-manifolds via circle-valued maps.

The study analyzes Dirac operators twisted by specific bundles, revealing their geometric and regularity properties.

Supposing that X is a Riemannian manifold, a Z/2 spinor on X is defined by a data set consisting of a closed set in X to be denoted by Z, a real line bundle over X-Z, and a nowhere zero section on X-Z of the tensor product of the real line bundle and a spinor bundle. The set Z and the spinor are jointly constrained by …

We estimate the dimensions of the spaces of holomorphic sections of certain line bundles to give improved lower bounds on the index of complex isotropic harmonic maps to complex projective space from the sphere and torus, and in some cases from higher genus surfaces.

Study finds how periodic surfaces can bend without stretching.

Minimal stretch factor for non-orientable surfaces is small.

The present article studies the class of Einstein-Hermitian harmonic maps of constant Kaehler angle from the projective line into quadrics. We provide a description of their moduli spaces up to image, and gauge-equivalence using the language of vector bundles and representation theory. It is shown that the dimension of…

The limit of energies of a sequence of harmonic maps as their annular domains approach the boundary of moduli space depends upon the boundary point approached. The infinite energy case is associated with limits of images containing ruled surfaces. The finite energy case yields a limit of images, under a suitable topolo…

The paper studies maximal stretch and Lipschitz maps on negatively curved manifolds.

Quadratic-time algorithm computes stretch factors and foliations for pseudo-Anosov mapping classes.

We explicitly construct pseudo-Anosov maps on the closed surface of genus $g$ with orientable foliations whose stretch factor $λ$ is a Salem number with algebraic degree $2g$. Using this result, we show that there is a pseudo-Anosov map whose stretch factor has algebraic degree $d$, for each positive even integer $d$ s…

Minimal stretch factors for certain pseudo-Anosov maps are bounded.

Study how large-scale flows align small-scale vortices in 3D Euler equations.

Study on solvable Lie groups with specific Weyl connections.

In this paper, we study the Galois conjugates of stretch factors of pseudo-Anosov elements of the mapping class group of a surface. We show that - except in low-complexity cases - these conjugates are dense in the complex plane. For this, we use Penner's construction of pseudo-Anosov mapping classes. As a consequence, …

Study connects group invariants through outer automorphisms and polynomial relations.

In this paper, we study the dimension of cohomology of semipositive line bundles over Hermitian manifolds, and obtain an asymptotic estimate for the dimension of the space of harmonic $(0,q)$-forms with values in high tensor powers of a semipositive line bundle when the fundamental estimate holds. As applications, we e…

In this paper we prove that on a complete smooth metric measure space with non-negative Bakry-Émery-Ricci curvature if the space of weighted L^2 harmonic one-forms is non-trivial then the weighted volume of the manifold is finite and universal cover of the manifold splits isometrically as the product of the real line w…

The paper examines properties of spherical Finsler metrics, proving their semi-C-reducibility and conditions for vanishing mean stretch curvature.

We determine the smallest stretch factor among pseudo-Anosov maps with an orientable invariant foliation on the closed nonorientable surfaces of genus 4, 5, 6, 7, 8, 10, 12, 14, 16, 18 and 20. We also determine the smallest stretch factor of an orientation-reversing pseudo-Anosov map with orientable invariant foliation…

Study reveals uniform difference in stretch factors between genus two handlebody group and outer automorphism group.

The motivation for this paper is to justify a remark of Thurston that the algebraic degree of stretch factors of pseudo-Anosov maps on a surface $S$ can be as high as the dimension of the Teichmüller space of $S$. In addition to proving this, we completely determine the set of possible algebraic degrees of pseudo-Anoso…

New geometric invariant limits the number of semi-arithmetic groups.