We will prove an upper bound for the volume of a minimal hypersurface in a closed Riemannian manifold conformally equivalent to a manifold with $\mathrm{Ric}>-(n-1)$. In the second part of the talk we will construct a sweepout of a closed 3-manifold with positive Ricci curvature by $1$-cycles of controlled length and prove a systolic inequality for such manifolds.

These are joint works with Parker Glynn-Adey (Toronto) and Xin Zhou (MIT)