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 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.
Cite
@article{arxiv.2507.23239,
title = {Min-max theory and minimal surfaces with prescribed genus},
author = {Adrian Chun-Pong Chu and Yangyang Li and Zhihan Wang},
journal= {arXiv preprint arXiv:2507.23239},
year = {2026}
}
Comments
The v1 manuscript has been split into two parts: the construction of a topologically optimal family is contained in v2, while the existence result for genus 2 minimal surfaces is strengthened and superseded by arXiv:2601.01736. v3: expanded the discussion of the main ideas in section 1.1