Cmc Doublings of Minimal Surfaces via Min-Max

TL;DR

This paper develops a min-max theoretical approach to construct constant mean curvature doublings of minimal surfaces with low index, extending previous gluing methods to new geometric settings.

Contribution

It introduces a min-max framework combined with the catenoid estimate to produce cmc doublings of minimal surfaces with index 0 or 1.

Findings

Constructed $ ext{ extepsilon}$-cmc doublings for small $ ext{ extepsilon}$

Extended previous gluing methods to min-max approach

Applicable to minimal surfaces with low index in 3-manifolds

Abstract

Let $\Sigma^2 \subset M^3$ be a minimal surface of index 0 or 1. Assume that a neighborhood of $\Sigma$ can be foliated by constant mean curvature (cmc) hypersurfaces. We use min-max theory and the catenoid estimate to construct $\varepsilon$-cmc doublings of $\Sigma$ for small $\varepsilon > 0$. Such cmc doublings were previously constructed for minimal hypersurfaces $\Sigma^n \subset M^{n+1}$ with $n+1\ge 4$ by Pacard and Sun using gluing methods.