An enumerative min-max theorem for minimal surfaces

Abstract

We prove an enumerative min-max theorem that relates the number of genus g minimal surfaces in 3-manifolds of positive Ricci curvature to topological properties of the set of embedded surfaces of genus $\le g$, possibly with finitely many singularities. This completes a central component of our program of using topological methods to enumerating minimal surfaces with prescribed genus. As an application, we show that every 3-sphere of positive Ricci curvature contains at least 4 embedded minimal surfaces of genus 2.

This paper supersedes the portion of arXiv:2507.23239v1 concerning the existence of genus 2 minimal surfaces