We prove a sharp Alexandrov-Fenchel-type inequality for star-shaped, strictly mean convex hypersurfaces in hyperbolic space. The argument uses two new monotone quantities for the inverse mean curvature flow. As an application we establish, in any dimension, an optimal Penrose inequality for asymptotically hyperbolic graphs carrying a minimal horizon, with the equality occurring if and only if the graph is an anti-de Sitter-Schwarzschild solution. This settles, for this class of initial data sets, the conjectured Penrose inequality for time-symmetric space-times with negative cosmological constant (joint work with Fred Girão - UFC).

We present two recent results on $r$-minimal hypersurfaces in Euclidean space, namely, an existence result for $r$-minimal hypersurfaces with ends of planar type via perturbative methods (joint work with J. de Lira and J. Silva) and a uniquess result, a la Schoen, for a certain class of two-ended $r$-minimal hypersurfaces (joint work with A. Souza).