Hyperelliptic surfaces are Loewner

Abstract

We prove that C. Loewner’s inequality for the torus is satisfied by conformal metrics on hyperelliptic surfaces $X$ as well. In genus 2, we first construct the Loewner loops on the (mildly singular) companion tori, locally isometric to $X$ away from Weierstrass points. The loops are then transplanted to $X$, and surgered to obtain a Loewner loop on $X$. In higher genus, we exploit M. Gromov’s area estimates for $\varepsilon$-regular metrics on $X$.