Mathematics Colloquium

Scalar curvature is a simple curvature invariant of a Riemannian manifold. Understanding its effects on the geometry and topology of a manifold has been a challenge for geometers. I will present some recent progress in this direction, including polyhedron comparison theorems of scalar curvature lower bounds, as well as non-existence of positive scalar curvature metrics on aspherical manifolds of dimensions 4 and 5 (as conjectured by Schoen-Yau and by Gromov). The solutions to these problems are based on constructing minimal surfaces and surfaces of prescribed mean curvature by calculus of variations. I will discuss a regularity question that naturally arises in the strategy.