The hyperbolic geometry of the symmetrized bidisc

Abstract

We solve the Caratheodory and Kobayashi extremal problems for the open symmetrized bidisc % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaGaem4raCucKbai-pacacidaqZ-meaabGaGaYaa08pc% eaGaiaiGmaan--hzaiacacidaqZ--vgacGaGaYaa08-FMbaaaeHbbj% xAHXgaiyaacaGFGaGaa4hiaKaaGiadaALG7bWEcWaGwjikaGIamaOv% dQha6PWaiaOvBaaaleacaALamaOvigdaXaqajaOvaOGamaOvgUcaRK% aaGiadaA1G6bGEkmacaA1gaaWcbGaGwjacaA1FYaaabKaGwbGccWaG% wjilaWIaiaOv+bcajaaicWaGwnOEaONcdGaGwTbaaSqaiaOvcWaGwH% ymaedabKaGwbqcaaIamaOvdQha6PWaiaOvBaaaleacaALaiaOv-jda% aeqcaAfajaaicWaGwjykaKIamaOvcQda6iadgALG8baFcWaGwnOEaO% NcdGaGwTbaaSqaiaOvcWaGwHymaedabKaGwbqcaaIamaOvcYha8jad% aALH8aapcWaGwHymaeJamaOvcYcaSiadgALG8baFcWaGwnOEaONcdG% aGwTbaaSqaiaOvcGaGw9NmaaqajaOvaKaaGiadaALG8baFcWaGwzip% aWJamaOvigdaXiadaALG9bqFcWaGwzOGIW8efv3ySLgznfgDOjdarC% qr1ngBPrginfgDObcv39gaiCqacWaGw1NaHmKcdGaGwXbaaSqajaOv% bGaGwjacaA1FYaaaaOGamaOvc6caUaaa!AF58! $$G\underline{\underline {def}} \{ (z_1 + z_2 , z_1 z_2 ):|z_1 |< 1,|z_2 |< 1\} \subset \mathbb{C}^2 .$$ We prove the equality of the Caratheodory and Kobayashi distances on G and describe the extremal functions for the two problems; they are rational of degree 1or 2.G is the first example of a non convexifiable domain for which the two distances coincide.