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 $\epsilon$-cmc doublings of $\Sigma$ for small $\epsilon >0$. 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.

32 pages. Comments are welcome