arXiv research

This paper explores infinite-dimensional Teichmüller spaces and their properties.

New asymmetric metric on Teichmüller space for surfaces.

Teichmuller solved the type problem for Riemann surfaces.

For each right-angled hexagon in the hyperbolic plane, we construct a one-parameter family of right-angled hexagons with a Lipschitz map between any two elements in this family, realizing the smallest Lipschitz constant in the homotopy class of this map relative to the boundary. As a consequence of this construction, w…

This is a mathematical commentary on Teichm{ü}ller's paper ``Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Fl{ä}chen'' (Determination of extremal quasiconformal maps of closed oriented Riemann surfaces). This paper is among the last (and may be the last one) that Teich…

Abstract notes on hyperbolic surfaces and Teichmüller spaces.

We provide a commentary on Teichm{ü}ller's paper "Extremale quasikonforme Abbildungen und quadratische Differentiale" (Extremal quasiconformal mappings of closed oriented Riemann surfaces), Abh. Preuss. Akad. Wiss., Math.-Naturw. Kl. 1940, No.22, 1-197 (1940). The paper is quoted in several works, although it was read …

We survey some major contributions to Riemann's moduli space and Teichm{ü}ller space. Our report has a historical character, but the stress is on the chain of mathematical ideas. We start with the introduction of Riemann surfaces, and we end with the discovery of some of the basic structures of Riemann's moduli space a…

This paper adapts Thurston's earthquake metric to Riemann surfaces with marked points.

Study of Teichmüller space geometry using infinitesimal and global methods.

We construct a new Riemannian metric on Goldman space $\mathcal{B}(S)$, the space of the equivalence classes of convex projective structures on the surface $S$, and then prove the new metric, as well as the metric of Darvishzadeh and Goldman, restricts to be the Weil-Petersson metric on Teichm$\ddot{u}$ller space, embe…

We review some ideas of Grothendieck and others on actions of the absolute Galois group Γ Q of Q (the automorphism group of the tower of finite extensions of Q), related to the geometry and topology of surfaces (mapping class groups, Teichm{ü}ller spaces and moduli spaces of Riemann surfaces). Grothendieck's motivation…

Grothendieck's Esquisse d'un programme is often referred to for the ideas it contains on dessins d'enfants, the Teichm{ü}ller tower, and the actions of the absolute Galois group on these objects or their etale fundamental groups. But this program contains several other important ideas. In particular, motivated by surfa…

Bonahon's method for compactifying Teichmüller space extended to non-compact surfaces.

We comment on Teichm{ü}ller 's paper ''Vollst{ä}ndige L{ö}sung einer Extremalaufgabe der quasikonformen Abbildung'' (Complete solution of an ex-tremal problem of the quasiconformal mapping),, published in 1941. In this paper, Teichm{ü}ller gives a proof of the existence of extremal quasiconformal mappings in the case o…

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 combinatorial structures for Teichmüller spaces with Thurston's metric are explored.

The paper describes geometric properties of Teichmüller space metrics.

Tissot's indicatrix theory is foundational for quasiconformal mappings.

The arc metric is an asymmetric metric on the Teichm{ü}ller space T(S) of a surface S with nonempty boundary. In this paper we study the relation between Thurston's compactification and the horofunction compactification of T(S) endowed with the arc metric. We prove that there is a natural homeomorphism between the two …

The paper explores families of Finsler metrics and their properties.

Commentary on Teichmüller's 1938 paper on conformal and quasiconformal mappings.

The zoology of singularities for Lorentzian manifold is slightly more complicated than for Riemannian manifolds. Our present work study Cauchy-compact globally hyperbolic singular flat spacetimes with extreme BTZ-like singular lines. We use the notion of BTZ-extension of a singular spacetime introduced in a previous pa…

This is a commentary on Teichm{ü}ller's paper Ein Verschiebungssatz der quasikonformen Abbildung (A displacement theorem of quasiconformal mapping), published in 1944. We explain in detail how Teichm{ü}ller solves the problem of finding the quasiconformal mapping from the unit disc to itself, sending 0 to a strictly ne…

The study bounds the number of closed geodesics in a specific orbit closure of surfaces.

Study on Teichmüller space of acute triangles using explicit calculations.

Analyzes convex structures in Teichmüller space unit tangent spheres.

This paper classifies commensurability of Deligne-Mostow lattices.

For a semisimple real Lie group $G$, we study topological properties of moduli spaces of polystable parabolic $G$-Higgs bundles over a Riemann surface with a divisor of finitely many distinct points. For a split real form of a complex simple Lie group, we compute the dimension of apparent parabolic Teichm{ü}ller compon…

Maps between acute triangles with minimal stretch found and studied.

Study on earthquake metric on Teichmüller space, proving properties and new completions.

We consider harmonic maps$u(z): \mathcal{X}_z\to N$ in a fixed homotopy class from Riemann surfaces $\mathcal{X}_z$ of genus $g\geq 2$ varying in the Teichmü{}ller space $\mathcal T$ to a Riemannian manifold $N$ with non-positive Hermitian sectional curvature. The energy function $E(z)=E(u(z))$ can be viewed as a funct…

Study on Blaschke locus with covariance metric properties.

In this note we will adapt Topping's $\mathcal{L}$-optimal transportation theory for Ricci flow to a more general situation, i.e. to a closed manifold $(M,g_{ij}(t))$ evolving by $\partial_tg_{ij}=-2S_{ij}$, where $S_{ij}$ is a symmetric tensor field of (2,0)-type on $M$. We extend some recent results of Topping, Lott …

Grötzsch's papers review progress in quasiconformal geometry.

Continuity of earthquake flow map transfers Teichmüller dynamics results.

Minkowski space is the local model of 3 dimensionnal flat spacetimes. Recent progress in the description of globally hyperbolic flat spacetimes showed strong link between Lorentzian geometry and Teichm{ü}ller space. We notice that Lorentzian generalisations of conical singularities are useful for the endeavours of desc…

There are certain families of words and word sequences (words in the generators of a two-generator group) that arise frequently in the Teichm{ü}ller theory of hyperbolic three-manifolds and Kleinian and Fuchsian groups and in the discreteness problem for two generator matrix groups. We survey some of the families of su…

For $g \ge 5$, we give a complete classification of the connected components of strata of abelian differentials over Teichmüller space, establishing an analogue of Kontsevich and Zorich's classification of their components over moduli space. Building off of work of the first author (arXiv:1901.05482), we find that the …

One-parameter hyperbolic planar motion was first studied by S. Y$\ddot{\texttt{u}}$ce and N. Kuruo$\tilde{\texttt{g}}$lu. Moreover, they analyzed the relationships between the absolute, relative and sliding velocities of one-parameter hyperbolic planar motion as well as the related pole curves, \cite{Yuc}. One-paramete…

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

We generalize a well-known existence and uniqueness result for equivariant harmonic maps due to Corlette, Donaldson, and Labourie to a non-compact infinite energy setting and analyze the asymptotic behaviour of the harmonic maps. When the relevant representation is Fuchsian and has hyperbolic monodromy, our constructio…

Classical Kleinian groups are discrete subgroups of isometries of H n. The well-known theory of Kleinian groups starts with the definition of their associated limit set in the boundary of H n , and includes the geometric properties of the quotient hyperbolic space. This approach, naively applied, fails in the Lorentzia…

Study of cuspidal edges on focal surfaces of regular surfaces.

New method glues Scherk surfaces into minimal surfaces, limiting possible outcomes.

Study on focal surfaces of lightcone framed surfaces in Lorentz-Minkowski 3-space.

Introduces hyperbolic generalized framed surfaces and their properties.

The paper studies special surfaces with a new type of support function.