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Weixu Su

Publications and source records attributed to Weixu Su.

At least 19 recordsLinked to original sources
arXiv2026

The Carath\'eodory metric on Teichm\"uller space of genus two surface

Let $\Tei_{g,n}$ be the Teichm\"uller space of Riemann surfaces of genus $g$ with $n$ punctures. It is conjectured that the Teichm\"uller and Carath\'{e}odory metrics agree on a Teichm\"{u}ller disk if and only if all the zeros of the corresponding holomorphic quadratic differential are of even order. The conjecture was proved by Gekhtman and Markovic for $\Tei_{0,5}\cong \Tei_{1,2}$. We confirm the conjecture for $\Tei_{2,0}\cong\Tei_{0,6}$.

math.CV↗
arXiv2026

Counting Saddle Connections on Hyperelliptic Translation Surfaces with a Slit

We consider saddle connections on a translation surface in a hyperelliptic connected component of a stratum that do not intersect the interior of a distinguished saddle connection. For this restricted set of saddle connections, we show that it satisfies an $L (\log L)^{d-2}$ growth rate, where $d$ is the complex dimension of the hyperelliptic stratum. The upper bound holds for all translation surfaces in the hyperelliptic stratum while the lower bound holds for almost every surface in the hyperelliptic stratum. The proof of the lower bound uses horocycle renormalization.

math.DS↗
arXiv2025

Volume of unit balls associated to quadratic differentials

Associated to a holomorphic quadratic differential is a unit ball of the measured lamination space. The Thurston volume of the unit ball defines a function on the moduli space. We show that the volume function is not proper and characterize when it tends to infinity. We prove that the volume function is $p$-integrable for any $0<p<1$.

math.CV↗
arXiv2024

Every closed surface of genus at least 18 is Loewner

In this paper, we obtain an improved upper bound involving the systole and area for the volume entropy of a Riemannian surface. As a result, we show that every orientable and closed Riemannian surface of genus $g\geq 18$ satisfies Loewner's systolic ratio inequality. We also show that every closed orientable and nonpositively curved Riemannnian surface of genus $g\geq 11$ satisfies Loewner's systolic ratio inequality.

math.DG↗
arXiv2023

Optimal geodesics for boundary points of the Gardiner-Masur compactification

The Gardiner-Masur compactification of Teichmüller space is homeomorphic to the horofunction compactification of the Teichmüller metric. Let $ξ$ and $η$ be a pair of boundary points in the Gardiner-Masur compactification that fill up the surface. We show that there is a unique Teichmüller geodesic which is optimal for the horofunctions corresponding to $ξ$ and $η$. In particular, when $ξ$ and $η$ are Busemann points that fill up the surface, the geodesic converges to $ξ$ in forward direction and to $η$ in backward direction. As an application, we show that if $\mathbf{G}_n$ is a sequence of Teichmüller geodesics passing through $X_n$ and $Y_n$ such that $X_n \to ξ$ and $Y_n \to η$, then $\mathbf{G}_n$ converges to a unique Teichmüller geodesic.

math.GT↗
arXiv2023

The geometry of the Thurston metric: a survey

This paper is a survey about the Thurston metric on the Teichmüller space. The central issue is the constructions of extremal Lipschitz maps between hyperbolic surfaces. We review several constructions, including the original work of Thurston. Coarse geometry and isometry rigidity of the Thurston metric, relation between the Thurston metric and the Thurston compactification are discussed. Some recent generalizations and developments of the Thurston metric are sketched.

math.GT↗
arXiv2021

Horospheres in Teichmüller space and mapping class group

We study the geometry of horospheres in Teichmüller space of Riemann surfaces of genus g with n punctures, where $3g-3+n\geq 2$. We show that every $C^1$-diffeomorphism of Teichmüller space to itself that preserves horospheres is an element of the extended mapping class group. Using the relation between horospheres and metric balls, we obtain a new proof of Royden's Theorem that the isometry group of the Teichmüller metric is the extended mapping class group.

math.GT↗
arXiv2021

Counting mapping class group orbits under shearing coordinates

Let $S_{g,n}$ be an oriented surface of genus $g$ with $n$ punctures, where $2g-2+n>0$ and $n>0$. Any ideal triangulation of $S_{g,n}$ induces a global parametrization of the Teichm\"uller space $\mathcal{T}_{g,n}$ called the shearing coordinates. We study the asymptotics of the number of the mapping class group orbits with respect to the standard Euclidean norm of the shearing coordinates. The result is based on the works of Mirzakhani.

math.GT↗
arXiv2020

Dominating surface-group representations into $\mathrm{PSL}_2 (\mathbb{C})$ in the relative representation variety

Let $ρ$ be a representation of the fundamental group of a punctured surface into $\mathrm{PSL}_2 (\mathbb{C})$ that is not Fuchsian. We prove that there exists a Fuchsian representation that strictly dominates $ρ$ in the simple length spectrum, and preserves the boundary lengths. This extends a result of Gueritaud-Kassel-Wolff to the case of $\mathrm{PSL}_2 (\mathbb{C})$-representations. Our proof involves straightening the pleated plane in $\mathbb{H}^3$ determined by the Fock-Goncharov coordinates of a framed representation, and applying strip-deformations.

math.GT↗
arXiv2019

Existence of closed geodesics through a regular point on translation surfaces

We show that on any translation surface, if a regular point is contained in a simple closed geodesic, then it is contained in infinitely many simple closed geodesics, whose directions are dense in the unit circle. Moreover, the set of points that are not contained in any simple closed geodesic is finite. We also construct explicit examples showing that such points exist. For a surface in any hyperelliptic component, we show that this finite exceptional set is actually empty. The proofs of our results use Apisa's classifications of periodic points and of $\GL(2,\R)$ orbit closures in hyperelliptic components, as well as a recent result of Eskin-Filip-Wright.

math.GT↗
arXiv2016

Variation of extremal length functions on Teichmuller space

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 functions along Weil-Petersson geodesics. We show that the extremal length of any measured foliation is a pluri-subharmonic function on Teichmuller space.

math.GT↗
arXiv2016

The horofunction compactification of the arc metric on Teichmüller space

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 compactifications.

math.GT↗
arXiv2015

Convergence of earthquake and horocycle paths to the boundary of Teichmüller space

We study the convergence of earthquake paths and horocycle paths in the Gardiner-Masur compactification of Teichmüller space. We show that an earthquake path directed by a uniquely ergodic or simple closed measured geodesic lamination converges to the Gardiner-Masur boundary. Using the embedding of flat metrics into the space of geodesic currents, we prove that a horocycle path in Teichmüller space, induced by a quadratic differential whose vertical measured foliation is unique ergodic, converges to the Gardiner-Masur boundary and to the Thurston boundary.

math.GT↗
arXiv2015

A commentary on Teichmüller's paper "Extremale quasikonforme Abbildungen und quadratische Differentiale"

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 by very few people. Some of the results it contains were rediscovered later on and published without any reference to Teichm{ü}ller.In this commentary, we highlight the main results and the main ideas contained in that paper and we describe some of the important developments they gave rise to.The final version of this paper, together with the English translation of Teichm{ü}ller's paper, will apper in Volume V of the \emph{Handbook of Teichm{ü}ller theory} (European Mathematical Society Publishing House, 2015).

math.GT↗
arXiv2015

Thurston's metric on Teichmüller space and the translation lengths of mapping classes

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 translation distances for mapping classes for their actions on Teichmüller spaces equipped with their arc metrics.

math.GT↗