Equivariant min-max theory

TL;DR

This paper develops an equivariant min-max theory to generate minimal surfaces in the 3-sphere, reproducing known examples and creating new families through doubling and desingularization techniques.

Contribution

It extends the equivariant min-max framework, producing both known and novel minimal surfaces in ^3 with specified genus and symmetry groups.

Findings

Reproduces many known minimal surfaces in ^3

Constructs new infinite families of minimal surfaces

Introduces methods for doubling and desingularization of varifolds

Abstract

We develop an equivariant min-max theory as proposed by Pitts-Rubinstein in 1988 and then show that it can produce many of the known minimal surfaces in $\mathbb{S}^3$ up to genus and symmetry group. We also produce several new infinite families of minimal surfaces in $\mathbb{S}^3$ proposed by Pitts-Rubinstein. These examples are doublings and desingularizations of stationary integral varifolds in $\mathbb{S}^3$.