Bangert’s question on codimension-one stable norms

Bangert asked whether a metric on a torus must be flat when every primitive codimension-one class has a foliation by minimal tori. The question now has a negative answer in every dimension $n \ge 3$.

August 2026 negative answer

• For every $n \ge 3$, there are nonflat metrics with the required minimal-foliation property, although their codimension-one stable unit sphere is not $C^2$.
• In dimension $3$, a smooth family starts at a unit-volume flat metric, keeps the same stable norm on $H_2(T^3,\mathbb{R})$, and becomes nonflat; its volume changes and can tend to $0$.
• A subfamily has the codimension-one stable norm of the cubic flat torus while its members are pairwise non-isometric, showing that stable-norm and volume data do not locally determine the metric.

Current status (as of August 2026): Bangert’s question is resolved negatively for every $n \ge 3$, while broader inverse problems for stable norms remain open.