Characterization of Gromov-Hausdorff-almost-flat tori

One of the goals of this talk is to establish a nonlinear analogue of a splitting map into a Euclidean space, as a harmonic map into a flat torus. Existence of such a map implies Gromov-Hausdorff closeness to a flat torus in any dimension. Furthermore, Gromov-Hausdorff closeness to a flat torus and an integral bound on the scalar curvature imply the existence of a harmonic splitting map. Combining these results with Stern's inequality yields a new Gromov-Hausdorff stability theorem for flat 3-tori.