  We establish a nonlinear analogue of a splitting map into a Euclidean space,
as a harmonic map into a flat torus. We prove that the 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 $r_M(x)$, the smallest eigenvalue of the Ricci tensor $\text{ric}_x$ in
$x$}, imply the existence of a harmonic splitting map. Combining these results
with Stern's inequality, we provide a new Gromov-Hausdorff stability theorem
for flat $3$-tori. The main tools we employ include the harmonic map heat flow,
Ricci flow, and both Ricci limits and RCD theories.
