Relative systoles in hyperelliptic translation surfaces

Abstract

In this paper we prove that the systole fonction on a connected component of area one translation surfaces admits a local maximum that is not a global maximum if and only if the connected component is not hyperelliptic.

Figures (4)

• Figure 1: The disk $D$ and the quadrilateral $\mathcal{Q}$ in 3 configurations.
• Figure 2: A global maximum with a closed shortest saddle connection $\gamma$ sastisfying $\mathrm{Ind}([\gamma])=0$
• Figure 3: Local but nonglobal maxima in $\mathcal{H}(2,0)$ and $\mathcal{H}(2,0,0)$
• Figure 4: Global maxima in $\mathcal{H}(2)$ and $\mathcal{H}(1,1)$

Theorems & Definitions (11)

• Theorem : Main Theorem
• Theorem 2.1: Kontsevich--Zorich
• Lemma 2.2
• Theorem 3.1
• Lemma 3.2
• proof : Proof of Theorem \ref{['th:cc:hyp']}
• proof : Proof of Lemma \ref{['lem:decrease:area:disk']}
• Theorem 4.1
• Lemma 4.2
• proof