Width, Ricci curvature and minimal hypersurfaces

Let $(M,g_0)$ be a closed Riemannian manifold of dimension $n$, for $3 \leq n \leq 7$,andnon-negativeRiccicurvature\mathrm{.}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$V$isthetotalvolumeof$(M,g)$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 $C_n V \left( sys_{n-1}(M)\right)^{-\frac{1}{n-1}} k ^ {\frac{1}{n-1}}$,where$V$denotesthevolumeof$(M,g_0)$and$sys_{n-1}(M)$ is the smallest volume of a non-trivial minimal hypersurface.