I shall describe an area comparison theorem for certain totally geodesic surfaces in 3-manifolds with lower bounds on the scalar curvature. This result is an optimal analogue of the Heintze-Karcher-Maeda area comparison theorem for minimal hypersurfaces in manifolds of non-negative Ricci curvature. I shall then show how this area comparison theorem provides a unified proof of three splitting and rigidity theorems for 3-manifolds with lower bounds on the scalar curvature that were first proved, independently, by Cai-Galloway, Bray-Brendle-Neves and Nunes. This is joint work with Mario Micallef. Finally, I shall address some natural higher dimensional generalisations of these area comparison and rigidity results.bstract

We consider generic measure-minimization problems with weak initial assumptions on the regularity of competitors (we do not suppose orientability nor even rectifiability). A subset of $\mathbf R^n$ is said to be minimal if its $d$-dimensional Hausdorff measure cannot be decreased by a deformation taken in a suitable homotopy class. The classic Plateau problem can be rewritten in these terms by finding a minimal set under deformations that only move a relatively compact subset of points of a given domain: in that case the boundary of the domain acts as a topological constraint. There are relatively few existence results under this setup, compared to classical approaches based upon differential geometry.
We provide a method to convert any minimizing sequence of the problem into a sequence of quasi-minimal sets which converges for the local Hausdorff distance by using a Federer-Flemming-like polyhedric approximation theorem by uniformly shaped polyhedrons. We show that the limit set is minimal, and in some cases this can be enough to control the topology. We give some examples of topological constraint for which it is possible to provide a complete existence result using this method.