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Ricci curvature and minimal hypersurfaces with large Betti numbers

Davi Maximo, Philipp Reiser, Daniele Semola

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.

Ricci curvature and minimal hypersurfaces with large Betti numbers

Abstract

In any dimension we construct a sequence of closed -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.

Paper Structure

This paper contains 4 sections, 13 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['mainthm']}
  4. From non-negative to positive Ricci curvature in the equivariant setting

Key Result

Theorem A

For $n+1\geq 4$, there exists a sequence of smooth, closed, $(n+1)$-dimensional, Riemannian manifolds $(M^{n+1}_k,g_k)$ with embedded, two-sided minimal hypersurfaces $\Sigma_k\subset M_k$ satisfying the following properties:

Theorems & Definitions (26)

  • Theorem A
  • Theorem B
  • Corollary A
  • Theorem C
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • ...and 16 more