Minimal surfaces in S3: a survey of recent results

Abstract

In this survey, we discuss various aspects of the minimal surface equation in the three-sphere S3. After recalling the basic definitions, we describe a family of immersed minimal tori with rotational symmetry. We then review the known examples of embedded minimal surfaces in S3. Besides the equator and the Clifford torus, these include the Lawson and Kapouleas-Yang examples, as well as a new family of examples found recently by Choe and Soret. We next discuss uniqueness theorems for minimal surfaces in S3, such as the work of Almgren on the genus 0 case, and our recent solution of Lawson’s conjecture for embedded minimal surfaces of genus 1. More generally, we show that any minimal surface of genus 1 which is Alexandrov immersed must be rotationally symmetric. We also discuss Urbano’s estimate for the Morse index of an embedded minimal surface and give an outline of the recent proof of the Willmore conjecture by Marques and Neves. Finally, we describe estimates for the first eigenvalue of the Laplacian on a minimal surface.