The extremal length systole of the Bolza surface

Extremal length is a conformal invariant that plays an important
role in Teichmueller theory. For each essential closed curve on a Riemann
surface, it furnishes a function on the Teichmueller space. The extremal
length systole of a Riemann surface is defined as the infimum of extremal
lengths of all essential closed curves. Its hyperbolic analogue is the
hyperbolic systole: the infimum of hyperbolic lengths of all essential
closed curves. While the latter has been studied profusely, the extremal
length systole remains widely unexplored. For example, it is known that in
genus 2, the hyperbolic systole has a unique global maximum: the Bolza
surface. In this talk we introduce the extremal length systole and show
that in genus two it attains a strict local maximum at the Bolza surface,
where it takes the value square root of 2. This is joint work with Maxime
Fortier Bourque and Franco Vargas Pallete.