Width, Ricci curvature, and minimal hypersurfaces

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 $C(n)V^{\frac{n-1}{n}}$, where $V$ is the total volume of $(M,g)$. 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 and dimension at most seven. We obtain an effective version of a theorem of F. C. 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 $C_n V ({\mathrm{sys}_{n-1}(M))}^{- \frac{1}{n-1}} k^{\frac{1}{n-1}}$, where $V$ denotes the volume of $(M,g_0)$ and $\mathrm{sys}_{n-1}(M)$ is the smallest volume of a non-trivial minimal hypersurface.