Pacific Dynamics Seminar: Didac Martinez Granado

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.