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The Carathéodory metric on Teichmüller space of genus two surface

Kejie Lin, Weixu Su

TL;DR

The paper resolves a key case of the long-standing question whether the Teichmüller metric $d_{\mathcal{T}}$ and Carathéodory metric $d_{\mathcal{C}}$ agree on Teichmüller disks by proving a holomorphic retract criterion for genus two, $\mathcal{T}_{2,0} \cong \mathcal{T}_{0,6}$. Building on Markovic’s criterion and orbit-closure reductions, the authors classify Jenkins–Strebel differentials on $S_{0,6}$, showing that many cases reduce to staircase types, for which they perform explicit Schwarz–Christoffel computations to obstruct holomorphic retractions. They prove that for staircase JS-differentials, no holomorphic retraction exists, and combine this with pullback arguments to confirm that $d_{\mathcal{T}} = d_{\mathcal{C}}$ on Teichmüller disks generated by quadratic differentials with all even zeros, or with a simple zero at a marked point, in $\mathcal{T}_{2,0}$. This confirms the GM2020 conjecture for genus two and has implications for the geometry of Teichmüller space, including consequences for rigidity and convexity questions in moduli theory. The methods achieve a concrete, computable obstruction via Schwarz–Christoffel maps and area expansions, potentially informing similar classifications in higher genus. All results are expressed with explicit $\mathcal{T}$–$\mathcal{C}$ criteria in terms of holomorphic retracts and orbit dynamics.

Abstract

Let $\Tei_{g,n}$ be the Teichmüller space of Riemann surfaces of genus $g$ with $n$ punctures. It is conjectured that the Teichmüller and Carathéodory metrics agree on a Teichmüller 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 $\Tei_{2,0}\cong\Tei_{0,6}$.

The Carathéodory metric on Teichmüller space of genus two surface

TL;DR

The paper resolves a key case of the long-standing question whether the Teichmüller metric and Carathéodory metric agree on Teichmüller disks by proving a holomorphic retract criterion for genus two, . Building on Markovic’s criterion and orbit-closure reductions, the authors classify Jenkins–Strebel differentials on , showing that many cases reduce to staircase types, for which they perform explicit Schwarz–Christoffel computations to obstruct holomorphic retractions. They prove that for staircase JS-differentials, no holomorphic retraction exists, and combine this with pullback arguments to confirm that on Teichmüller disks generated by quadratic differentials with all even zeros, or with a simple zero at a marked point, in . This confirms the GM2020 conjecture for genus two and has implications for the geometry of Teichmüller space, including consequences for rigidity and convexity questions in moduli theory. The methods achieve a concrete, computable obstruction via Schwarz–Christoffel maps and area expansions, potentially informing similar classifications in higher genus. All results are expressed with explicit – criteria in terms of holomorphic retracts and orbit dynamics.

Abstract

Let be the Teichmüller space of Riemann surfaces of genus with punctures. It is conjectured that the Teichmüller and Carathéodory metrics agree on a Teichmüller 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 . We confirm the conjecture for .
Paper Structure (15 sections, 15 theorems, 114 equations, 36 figures)

This paper contains 15 sections, 15 theorems, 114 equations, 36 figures.

Table of Contents

  1. Introduction
  2. Teichmüller disks and Orbit closures
  3. Teichmüller disks and holomorphic retracts
  4. Orbit closures
  5. Markovic's criterion
  6. Classification of Jenkins-Strebel differentials on $S_{0,6}$
  7. Case (I): $\phi \in \mathcal{Q}(1^2, -1^6)$.
  8. Case (II): $\mathcal{Q}(2, -1^6)$.
  9. Case (III): $\mathcal{Q}(1, -1^5)$.
  10. Case (IV): $\mathcal{Q}(-1^4)$.
  11. Conclusion
  12. Staircase surfaces
  13. Holomorphic retraction
  14. Computation of area.
  15. Proof of Theorem \ref{['thm:typeA']}

Key Result

Theorem 1.2

Let $\tau^{\phi}$ be a Teichmüller disk in $\mathcal{T}_{2,0}$. The Teichmüller metric and Carathéodory metric agree on $\tau^{\phi}$ if and only if $\phi$ is a quadratic differential all of whose zeros are of even order.

Figures (36)

  • Figure 1: The double of the polygon is called a staircase.
  • Figure 2: Case (I-1): The crosses stand for simple poles and the black dots stand for zeros. The Jenkins-Strebel differential $\phi$ is decomposed by the critical graph into three cylinders.
  • Figure 3: The three types of polygon corresponding to types (a), (b) and (c). The one of type (a) is a staircase-shaped polygon.
  • Figure 4: The steps to deform a pillowcase surface of type (b) into a staircase. The dashed lines in color red denote vertical saddle connections.
  • Figure 5: The step to deform a pillowcase of type (c) with two vertical cylinders into a staircase surface.
  • ...and 31 more figures

Theorems & Definitions (30)

  • Remark 1
  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2
  • Definition 2.1: Holomorphic retract
  • Lemma 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Remark 3
  • ...and 20 more