## Analysis Seminar

Let M be a closed hyperbolic 3-manifold, and assume that M contains a closed totally geodesic surface S. Then if g is a Riemannian metric on M with scalar curvature greater than or equal to −6, we show that the area in g of any surface homotopic to S is greater than or equal to the area of S in the hyperbolic metric, with equality only if g is isometric to the hyperbolic metric. We also consider a functional introduced by Calegari-Marques-Neves that is defined by an asymptotic count of minimal surfaces in (M, g). We show this functional to be uniquely maximized, over all metrics of scalar curvature greater than or equal to −6, by the hyperbolic metric. Our proofs use the Ricci flow with surgery. This talk is based on joint work with Andre Neves.