Ricci curvature and minimal hypersurfaces with large Betti numbers

TL;DR

This paper constructs high-dimensional Riemannian manifolds with positive Ricci curvature that contain minimal hypersurfaces with unbounded Betti numbers, revealing new geometric-topological phenomena.

Contribution

It provides explicit examples of manifolds with positive Ricci curvature hosting minimal hypersurfaces with unbounded Betti numbers, advancing understanding of curvature and topology interactions.

Findings

Constructed sequences of manifolds with positive Ricci curvature and unbounded Betti numbers.

Minimal hypersurfaces in these manifolds have Morse index one.

First Betti numbers of hypersurfaces are not uniformly bounded.

Abstract

In any dimension $n+1\ge 4$ we construct a sequence of closed $(n+1)$-dimensional Riemannian manifolds with positive Ricci curvature admitting embedded two-sided minimal hypersurfaces such that the following hold: (i) any such hypersurface has Morse index one; (ii) the first Betti numbers of the hypsersurfaces are not uniformly bounded along the sequence.