Minimal surfaces with arbitrary genus in 3-spheres of positive Ricci curvature

TL;DR

This paper explores the topology of singular surfaces in 3-spheres with positive Ricci curvature and proves the existence of embedded minimal surfaces of arbitrary genus with controlled area.

Contribution

It introduces new topological insights into singular surfaces and demonstrates the existence of minimal surfaces of any genus within positively curved 3-spheres.

Findings

Existence of genus g minimal surfaces in 3-spheres of positive Ricci curvature.

Bound on the area of these minimal surfaces relative to the first Simon-Smith width.

Topological structure of singular surfaces in the 3-sphere.

Abstract

We describe some topological structure in the set of all surfaces with finitely many singularities in the 3-sphere. As an application, we prove that every Riemannian 3-sphere of positive Ricci curvature contains, for every g, a genus g embedded minimal surface with area at most twice the first Simon-Smith width of the ambient 3-sphere.