A Survey of Convexity Estimates and Singularities for Mean Curvature Flow with Surgery

Abstract

In this survey we aim to introduce the basics of Mean Curvature Flow and detail the programme of Mean Curvature Flow with surgery developed by Huisken and Sinestrari over four seminal papers, [29],[28],[30] and [8]. We divide the survey into 3 chapters, the rst being an introduction to the Mean Curvature Flow. This is followed by the convexity estimates that were developed by Huisken and Sinestrari, which allow us to understand the singularities that form under Mean Curvature Flow. We also present an alternative proof of the convexity estimates for compact mean-convex Mean Curvature Flow that avoids the use of induction on symmetric functions and is based o the proof of Ben Andrews, James Mccoy and Mat Langford in [5]. In chapter 3 we use the convexity estimates for the 2-convex case to discuss the surgery procedure. We also compare and contrast aspects of the Ricci Flow and Hamilton's surgery programme for 4 dimensional manifolds with Positive Isotropic Curvature undergoing Ricci Flow which motivated Huisken and Sinestrari's surgery procedure. In this chapter we explain too the added technicality that was required to extend Mean Curvature Flow with surgery to dimension n = 2, achieved in [8], and we relate it to Perelman's surgery programme that extended Ricci Flow to 3-manifolds. Finally, we conclude with some topological applications and a discussion of further open problems. Show more