Qualitative and quantitative estimates for minimal hypersurfaces with bounded index and area

TL;DR

This paper establishes qualitative and quantitative estimates for the total curvature of closed minimal hypersurfaces with bounded index and area in low-dimensional Riemannian manifolds, linking curvature, topology, and limits.

Contribution

It provides new qualitative estimates on total curvature based on index and area, including quantization results and topological implications for minimal hypersurfaces.

Findings

Total curvature is quantized in the limit sequence.

Bounded index and area imply qualitative control on topology.

Limit surfaces' total curvature influences the sequence's curvature behavior.

Abstract

We prove qualitative estimates on the total curvature of closed minimal hypersurfaces in closed Riemannian manifolds in terms of their index and area, restricting to the case where the hypersurface has dimension less than seven. In particular, we prove that if we are given a sequence of closed minimal hypersurfaces of bounded area and index, the total curvature along the sequence is quantised in terms of the total curvature of some limit surface, plus a sum of total curvatures of complete properly embedded minimal hypersurfaces in Euclidean space - all of which are finite. Thus, we obtain qualitative control on the topology of minimal hypersurfaces in terms of index and area as a corollary.