I shall first review well-known results of Simons and Schoen- Yau on stable minimal hypersurfaces in manifolds with lower curvature bounds. Then I shall describe some joint work with Vlad Moraru on an area comparison result for certain totally geodesic surfaces in 3-manifolds with a lower bound on the scalar curvature. This result is a generalisation of a comparison theorem of Heintze-Karcher for minimal hypersurfaces in manifolds of nonnegative Ricci curvature. Our assumptions on the ambient 3-manifold are weaker than those of Heintze- Karcher but the assumptions on the surface are considerably more restrictive. I will then show how our comparison theorem provides a unified proof of various splitting theorems for 3-manifolds with lower bounds on the scalar curvature that were first proved separately by Cai-Galloway, Bray-Brendle-Neves and Nunes.

In this talk, I shall challenge the popular belief that Douglas arrived at his mysterious functional for solving the Plateau Problem by direct consideration of Dirichlet's integral and its relation to the area functional. I shall describe how, by looking at abstracts of Jesse Douglas in the Bulletin of the American Mathematical Society, I have been able to infer how Douglas MAY have arrived at his functional. Douglas was awarded one of the first Fields Medals for his work on the Plateau problem. I shall talk about some of the amusing aspects of the Fields Medal ceremony at which Douglas was awarded his prize. This is a joint work with the mathematical historian Jeremy Gray.