  Work of D. Stern and Bray-Kazaras-Khuri-Stern provide differential-geometric
identities which relate the scalar curvature of Riemannian 3-manifolds to
global invariants in terms of harmonic functions. These quantitative formulas
are useful for stability results and show promise for more applications of this
type. In this paper, we analyze harmonic maps to flat model spaces in order to
address conjectures concerning the geometric stability of the positive mass
theorem and the Geroch conjecture. By imposing integral Ricci curvature and
isoperimetric bounds, we leverage the previously mentioned formulas to
establish strong control on these harmonic maps. When the mass of an
asymptotically flat manifold is sufficiently small or when a Riemannian torus
has almost non-negative scalar curvature, we upgrade the maps to
diffeomorphisms and give quantitative H\"older closeness to the model spaces.
