Minimal hypersurfaces with bounded index

TL;DR

This paper provides a detailed local description of how sequences of closed embedded minimal hypersurfaces with bounded index degenerate in low-dimensional Riemannian manifolds, revealing their qualitative stability-like behavior.

Contribution

It establishes a structural theorem that characterizes the degeneration of minimal hypersurfaces with bounded index, leading to new compactness and finiteness results.

Findings

Hypersurfaces with bounded index behave like stable hypersurfaces locally.

The paper proves a precise local degeneration picture for these hypersurfaces.

Several new compactness and finiteness theorems are derived.

Abstract

We prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index (and volume if the ambient dimension is greater than three) in a Riemannian manifold of dimension at most seven, can degenerate. Loosely speaking, our results show that embedded minimal hypersurfaces with bounded index behave qualitatively like embedded stable minimal hypersurfaces, up to controlled errors. Several compactness/finiteness theorems follows our local picture.