Cmc Doublings of Minimal Surfaces via Min-Max

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.