Appears in collection : Not Only Scalar Curvature Seminar

In this talk I will present a proof of Gromov's conjecture on the total mean curvature of fill-ins in various cases. Our methods are based on surgery to reduce the statement to fill-ins of spheres, which can be treated by instances of the positive mass theorem. The case of spin fill-ins builds on a recent positive mass theorem with creases by Kazaras-Khuri-Lin. As another application of this, I will present a lock principle for scalar curvature. This gives a positive mass theorem for singular manifolds, where some components of the singular set may be mean-concave, provided that other components of the singular set are sufficiently mean-convex. This is based on joint work with Bernhard Hanke and Sven Hirsch.