Rigidity and non-rigidity of the stable norm on $T^n$

We show that the stable norm of flat metrics on $H_d(T^n,\mathbb{R})$ is locally rigid if $1\le d<n-1$ and locally rigid among metrics of the same volume if $d=n-1$. We also show that the stable norm on $H_2(T^3,\mathbb{R})$ is not locally rigid. As applications, we answer negatively a question raised by Bangert in his ICM address, prove local rigidity of the marked $k$-area spectrum of flat metrics for $1\le k\le n-2$, and prove a local rigidity result for the volume spectrum of flat metrics on $T^n$.