Area and boundary length of surfaces diffeomorphic to annuli

TL;DR

This paper provides an extrinsic proof for a statement in Perelman's work on Ricci flow extinction time, using the co-area formula, with potential for higher-dimensional generalization.

Contribution

It introduces a new extrinsic proof technique for Ricci flow results, differing from previous intrinsic methods, and suggests broader applicability.

Findings

Proof confirms a key statement in Ricci flow theory

Uses co-area formula instead of Gauss-Bonnet theorem

Potential for extending to higher dimensions

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.