[Paper Review] Singularities of quadratic differentials and extremal Teichmüller mappings defined by Dehn twists

Chaohui Zhang|ArXiv.org|Aug 17, 2007

Geometric and Algebraic Topology4 references3 citations

TL;DR

This paper investigates the location of zeros of holomorphic quadratic differentials associated with extremal Teichmüller mappings defined by Dehn twists on Riemann surfaces. Using geometric and quasiconformal techniques, it proves that non-puncture zeros of the differential must lie outside the closures of once-punctured disk components of the surface cut by two filling families of geodesics, and each disk component contains at most one zero. This leads to a sharp upper bound on the number of distinct zeros and poles of the differential.

ABSTRACT

Let $S$ be a Riemann surface of type $(p,n)$ with $3p-3+n>0$. Let $ω$ be a pseudo-Anosov map of $S$ that is obtained from Dehn twists along two families $\{A,B\}$ of simple closed geodesics that fill $S$. Then $ω$ can be realized as an extremal Teichmüller mapping on a surface of type $(p,n)$ which is also denoted by $S$. Let $ϕ$ be the corresponding holomorphic quadratic differential on $S$. In this paper, we compare the locations of some distinguished points on $S$ in the $ϕ$-flat metric to their locations with respect to the complete hyperbolic metric. More precisely, we show that all possible non-puncture zeros of $ϕ$ must stay away from all closures of once punctured disk components of $S\backslash \{A, B\}$, and the closure of each disk component of $S\backslash \{A, B\}$ contains at most one zero of $ϕ$. As a consequence of the result, we assert that the number of distinct zeros and poles of $ϕ$ is less than or equal to the number of components of $S\backslash \{A, B\}$.

Motivation & Objective

• To determine the geometric distribution of zeros of holomorphic quadratic differentials associated with pseudo-Anosov maps arising from Dehn twists.
• To analyze how the flat metric defined by the quadratic differential interacts with the hyperbolic metric and the topology of the surface after cutting along filling geodesic families.
• To establish constraints on the number and location of zeros and poles of the differential in terms of the components of the surface minus the geodesic families.
• To prove that the number of distinct zeros and poles of the differential is bounded by the number of components in the decomposition of the surface by the geodesic families.

Proposed method

• Lifts the Dehn twist word to a quasiconformal map on the universal cover of the surface, using the Bers fiber space construction.
• Applies the Ahlfors-Bers theory to associate a Teichmüller geodesic in the Teichmüller space of the surface.
• Uses the existence of a fixed point under the lifted map to derive a contradiction if a non-puncture zero lies in a once-punctured disk component.
• Employs a Bers isomorphism to embed a Teichmüller geodesic into the Teichmüller space of a punctured surface, leading to a contradiction via non-hyperbolic modular invariance.
• Applies the Riemann-Roch theorem to ensure existence of zeros on compactified surfaces.
• Uses the uniqueness of invariant geodesics under hyperbolic transformations to show that each disk component contains at most one zero.

Experimental results

Research questions

• RQ1 Where can non-puncture zeros of the holomorphic quadratic differential lie relative to the decomposition of the surface by two filling families of geodesics?
• RQ2 Can a disk component in the complement of the geodesic families contain more than one zero of the differential?
• RQ3 What is the maximum number of distinct zeros and poles the differential can have in relation to the number of components in the surface decomposition?
• RQ4 Does the presence of once-punctured disk components force all zeros to be at the punctures?
• RQ5 Can a non-puncture zero exist in a once-punctured disk component without violating the extremality or geometric structure of the Teichmüller mapping?

Key findings

• All non-puncture zeros of the holomorphic quadratic differential φ must lie in the complement of the closures of the once-punctured disk components of S\{A,B}.
• Each disk component in S\{A,B} contains at most one zero of φ.
• If S\{A,B} consists only of once-punctured disk components, then all zeros of φ are necessarily at the punctures of S.
• The total number of distinct zeros and poles of φ is at most the number of components in S\{A,B}.
• The proof relies on a contradiction derived from constructing a non-hyperbolic modular transformation that preserves a Teichmüller geodesic in a punctured surface, violating the pseudo-Anosov nature of the mapping.
• The method does not determine the order of individual zeros, only their location and count.