Appears in collection : Not Only Scalar Curvature Seminar

We use generalized minimal or capillary hypersurfaces to study comparison theorems of certain class of positively curved manifolds (with boundary); this is also called Gromov's $\mu$-bubble method. We apply these results to obtain useful geometric bounds such as Urysohn width, bandwidth estimates, and study rigidity of (free boundary) minimal hypersurfaces.