Area and boundary length of surfaces diffeomorphic to annuli

Abstract: In this paper, we give a proof to a statement in Perelman's paper for finite extinction time of Ricci flow. Our proof draws on different techniques from the one given in Morgan-Tian's exposition and is extrinsic in nature, which relies on the co-area formula instead of the Gauss-Bonnet theorem, and is potentially generalizable to higher dimensions.