An enumerative min-max theorem for minimal surfaces

TL;DR

This paper establishes a new min-max theorem linking the count of genus g minimal surfaces in positively curved 3-manifolds to topological properties of embedded surfaces, advancing the enumeration of minimal surfaces with fixed genus.

Contribution

It introduces an enumerative min-max theorem connecting minimal surface counts to topological features, completing a key part of the topological enumeration program.

Findings

Proves a min-max theorem relating minimal surface counts to topology.

Shows every 3-sphere with positive Ricci curvature has at least 4 genus 2 minimal surfaces.

Advances the topological approach to enumerating 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 $\leq g$, 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.