Width, Ricci curvature and minimal hypersurfaces

TL;DR

This paper establishes bounds on the volume of minimal hypersurfaces in conformally changed manifolds with non-negative Ricci curvature and proves the existence of infinitely many minimal hypersurfaces with controlled volume in positively Ricci curved manifolds.

Contribution

It provides new volume bounds for minimal hypersurfaces under conformal changes and offers an effective version of the existence of infinitely many minimal hypersurfaces in positively Ricci curved manifolds.

Findings

Existence of minimal hypersurfaces with volume bounds depending on total volume and conformal factor.

Effective bounds for multiple minimal hypersurfaces in positively Ricci curved manifolds.

Volume estimates involving the systole of the manifold.

Abstract

Let $(M,g_0)$ be a closed Riemannian manifold of dimension $n$, for $3 \leq n \leq 7$, and non-negative Ricci curvature. Let $g = \phi^2 g_0$ be a metric in the conformal class of $g_0$. We show that there exists a smooth closed embedded minimal hypersurface in $(M,g)$ of volume bounded by $C V^{\frac{n-1}{n}}$, where $V$ is the total volume of $(M,g)$ and $C$ is a constant that depends only on $n$. When $Ric(M,g_0) \geq -(n-1)$ we obtain a similar bound with constant $C$ depending only on $n$ and the volume of $(M,g_0)$. Our second result concerns manifolds $(M,g)$ of positive Ricci curvature. We obtain an effective version of a theorem of F. Coda Marques and A. Neves on the existence of infinitely many minimal hypersurfaces on $(M,g)$. We show that for any such manifold there exists $k$ minimal hypersurfaces of volume at most $C_n V \left( sys_{n-1}(M)\right)^{-\frac{1}{n-1}} k ^ {\frac{1}{n-1}}$, where $V$ denotes the volume of $(M,g_0)$ and $sys_{n-1}(M)$ is the smallest volume of a non-trivial minimal hypersurface.