Gromov–Hausdorff stability of tori under Ricci and integral scalar curvature bounds

Abstract

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 $r_M\left(x\right)$, the smallest eigenvalue of the Ricci tensor ${ric}_x$ 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.