POSITIVE SCALAR CURVATURE ON MANIFOLDS WITH BOUNDARY

Since work of Gromov and Lawson around 1980, we have known (under favorable circumstances) necessary and sufficient conditions for a closed manifold to admit a Riemannian metric of positive scalar curvature, but not much was known about analogous results for manifolds with boundary (and suitable boundary conditions). In joint work with Shmuel Weinberger of the University of Chicago, we give necessary and sufficient conditions in many cases for compact manifolds with non-empty boundary to admit: (a) a positive scalar curvature metric which is a product metric in a neighborhood of the boundary or (b) a positive scalar curvature metric with positive mean curvature on the boundary. The proof of the necessary conditions in some cases uses noncommutative geometry methods.