Algebraic intersection for translation surfaces in the stratum ℋ(2)

Smail Cheboui,Arezki Kessi,Daniel Massart

doi:10.5802/crmath.153

Abstract

We study the quantity $\mbox{KVol}$ defined as the supremum, over all pairs of closed curves, of their algebraic intersection, divided by the product of their lengths, times the area of the surface. The surfaces we consider live in the stratum $\mathcal{H}(2)$ of translation surfaces of genus $2$, with one conical point. We provide an explicit sequence $L(n,n)$ of surfaces such that $\mbox{KVol}(L(n,n)) \longrightarrow 2$ when $n$ goes to infinity, $2$ being the conjectured infimum for $\mbox{KVol}$ over $\mathcal{H}(2)$.

Highlights

Let X be a closed surface, that is, a compact, connected manifold of dimension 2, without boundary
Let us assume X is endowed with a Riemannian metric g
We have reasons to believe that KVol behaves differently in H (2), both from the systolic volume in H (2), and from KVol itself in the moduli space of hyperbolic surfaces of genus 2; that is, KVol does not have a minimum over H (2)

Summary

Introduction

Let X be a closed surface, that is, a compact, connected manifold of dimension 2, without boundary. We denote Vol(X , g ) the Riemannian volume of X with respect to the metric g , and for any piecewise smooth closed curve KVol(X , g ) = Vol(X , g ) sup α,β lg (α)lg (β) where the supremum ranges over all piecewise smooth closed curves α and β in X .

Results

Conclusion