On extremal discs inside compact hyperbolic surfaces

Abstract

It has been proved by C. Bavard that the radius of a disc isometrically embedded in a compact hyperbolic surface of genus g is bounded by R g = cosh ⁡ − 1 ( 1 2 sin ⁡ π 12 g - 6 ) , and that surfaces containing discs of such extremal radius are found in every genus. By constructing explicit surfaces of genus 2, he has also shown that extremal discs may or may not be unique. Here we show that a compact surface of genus g > 3 has at most one embedded extremal disc.