There is a special relationship between the Jacobi operator and the ambient scalar curvature operator, which we'll exploit. First, I'll talk about a "cut-and-fill" technique that simplifies 3-manifolds of nonnegative scalar curvature. This was used in the study of a priori L^1 estimates for the boundary mean curvature of mean-convex fill-ins with nonnegative scalar curvature, and to generalize Brown-York mass. Second, I'll talk about an extremal bending technique that lets us compute the Bartnik mass of apparent horizons. Parts of this talk reflect work done jointly with R. Schoen/P. Miao.