Min-max theory and minimal surfaces with prescribed genus

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 surfaces with possibly 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.

arXiv commentv1 is superseded by the current version and arXiv:2601.01736. v3: Added a proof sketch. v4: Minor revision for Prop 5.7