Existence of minimal surfaces of arbitrary large Morse index

TL;DR

This paper proves that in certain 3-manifolds, there exist minimal surfaces with arbitrarily large Morse index, advancing understanding of the relationship between geometry and minimal surface complexity.

Contribution

It demonstrates the existence of minimal surfaces with unbounded Morse index in closed 3-manifolds with positive Ricci curvature, confirming part of Marques and Neves' conjecture.

Findings

Existence of minimal surfaces with arbitrarily large Morse index

Analysis of lamination structures of minimal surface limits

Partial confirmation of Marques and Neves' conjecture

Abstract

We show that in a closed 3-manifold with a generic metric of positive Ricci curvature, there are minimal surfaces of arbitrary large Morse index, which partially confirms a conjecture by F. Marques and A. Neves. We prove this by analyzing the lamination structure of the limit of minimal surfaces with bounded Morse index.