Stable norm and volume rigidity on standard tori

Determine whether every smooth Riemannian metric on the standard marked torus ^n whose codimension-one stable norm is the Euclidean norm on all of H_{n-1}(^n,) and whose total volume is 1 must be isometric to a flat metric by a diffeomorphism isotopic to the identity.

Background

The paper constructs nonflat metrics on the three-torus whose codimension-one stable norm agrees exactly with that of the unit Euclidean flat torus. These examples show that the stable norm alone does not characterize flatness, but their volumes are strictly less than 1, so they do not resolve whether adding the unit-volume condition restores rigidity.

The question asks whether equality of the full Euclidean codimension-one stable norm together with unit total volume forces flatness globally among all smooth metrics on the standard marked torus. The paper proves an affirmative result only within its cohomogeneity-one metric class, while noting that local rigidity under these data was established independently elsewhere.

References

Let $g$ be a smooth Riemannian metric on the standard marked torus $n$. Suppose that

|r_{a_1,a_2,\ldots, a_n}|_{st,g}=\sqrt{a_12+a_22+\cdots +a_n2}

for every $(a_1, \ldots, a_n)\in H_{n-1}(n,)\congn$, and that

\operatorname{Vol}(n,g)=1.

Must $g$ be isometric to a flat metric by a diffeomorphism isotopic to the identity?

— Minimal foliations, codimension-one stable norms, and a question of Bangert  (2608.18428 - Nguyen, 19 Aug 2026) in Question 1, Section 4.1 ("Stable norm and volume rigidity on ^n")

For $n\ge4$, does there exist a smooth nonflat metric on the standard torus $n$ whose stable norm on $H_{n-1}(n,)$ is exactly

(a_1,\ldots,a_n)\longmapsto \sqrt{a_12+\cdots+a_n2}

under the standard integral identification?

— Minimal foliations, codimension-one stable norms, and a question of Bangert  (2608.18428 - Nguyen, 19 Aug 2026) in Question 2, Section 4.2 ("Higher-dimensional prescribed Euclidean stable norm")