Minimal surfaces in S^3: a survey of recent results

TL;DR

This survey reviews recent advances in the study of minimal surfaces in the three-sphere, including classifications, examples, uniqueness theorems, and connections to the Willmore conjecture.

Contribution

It compiles and discusses recent results on minimal surfaces in S^3, highlighting new examples, classification theorems, and solutions to longstanding conjectures.

Findings

Classification of minimal tori with rotational symmetry

Proof of Lawson's conjecture for genus 1 minimal surfaces

Outline of the proof of the Willmore conjecture

Abstract

In this survey, we discuss various aspects of the minimal surface equation in the three-sphere S^3. 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 S^3. 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 S^3, 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.