CMC hypersurfaces with bounded Morse index

TL;DR

This paper develops a bubble-compactness theory for constant mean curvature hypersurfaces with bounded Morse index, showing convergence properties and bounding their geometric complexity in low-dimensional Riemannian manifolds.

Contribution

It introduces a new bubble-compactness framework for embedded CMC hypersurfaces with bounded index and area, and establishes bounds on their genus and area.

Findings

Convergence occurs with multiplicity one.

All minimal blow-ups are catenoids.

Bounds on area and genus of CMC surfaces with bounded index.

Abstract

We develop a bubble-compactness theory for embedded CMC hypersurfaces with bounded index and area inside closed Riemannian manifolds in low dimensions. In particular we show that convergence always occurs with multiplicity one, which implies that the minimal blow-ups (bubbles) are all catenoids. We also provide bounds on the area of separating CMC surfaces of bounded (Morse) index and use this, together with the previous results, to bound their genus.