Fill-In Problems with Scalar Curvature Constraints

A central problem in differential geometry is understanding how the geometry of a boundary determines the geometry of its interior. Gromov's fill-in problem suggests that when a closed Riemannian manifold is filled with a region of large curvature, the extrinsic curvature of the boundary must be bounded above in some sense. The fill-in problem, particularly in the context of scalar curvature, is closely related to certain notions of quasi-local mass in general relativity. In this talk, I will discuss recent progress on the scalar curvature fill-in problem for the torus, which builds upon Brendle and Hung's resolution of the Horowitz-Myers conjecture.