Min-max theory and minimal surfaces with prescribed genus

TL;DR

This paper develops a min-max theorem to produce minimal surfaces with specific genus in 3-manifolds with positive Ricci curvature, advancing topological methods for constructing such surfaces.

Contribution

It introduces a general min-max framework for minimal surfaces with prescribed genus in positively curved 3-manifolds, including deformation techniques for families of surfaces.

Findings

Established a min-max theorem for minimal surfaces with prescribed genus.

Proved deformation of surface families into topologically optimal families.

Facilitated construction of multiple minimal surfaces in 3-spheres.

Abstract

We establish a general min-max type theorem that produces minimal surfaces with prescribed genus in 3-manifolds with positive Ricci curvature. An important intermediate step is to show that, in a generic metric with positive Ricci curvature, any family of smooth embedded surfaces, possibly with finitely many singularities, can be deformed into a certain topologically optimal family. Results in this paper will be crucial to our program on the construction of multiple minimal surfaces with prescribed genus in 3-spheres via topological methods.