Width, Ricci curvature and minimal hypersurfaces

Abstract

Let $(M,g_0)$ be a closed Riemannian manifold of dimension $n$, for $3\le n\le 7$, and non-negative Ricci curvature. Let $g={\varphi }^2{g}_0$ 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 $Ric(M,g_0)\ge -(n-1)$ 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 $C_{n}V{\left(sy{s}_{n-1}(M)\right)}^{-\frac{1}{n-1}}{k}^{\frac{1}{n-1}}$, where $V$ denotes the volume of $(M,g_0)$ and $sy{s}_{n-1}(M)$ is the smallest volume of a non-trivial minimal hypersurface.

19 pages, 1 figure. Improved exposition, minor corrections. To appear in J. Differential Geom