Minimal isometric immersions of flat n-tori into spheres

In 1985, Bryant stated that a flat $2$-torus admits a minimal isometric immersion into some round sphere if and only if a certain rationality condition is satisfied. We show that the rationality criterion is no longer a necessary, but a sufficient condition for a flat $n$-torus to admit minimal isometric immersions into spheres. We also derive an upper bound for the algebraic irrationality degree of such immersion. This bound is sharp and equals 4 if $n=3$, and explicit embedded examples are provided respectively for each possible degree. A non-homogeneous example is also presented to show that the minimal isometric immersion of flat $n$-tori is no longer necessarily homogeneous when $n\ge 3$. Moreover, we establish a deformation theorem that every flat n-torus admitting a minimal isometric spherical immersion can be isometrically, minimally and homogeneously immersed into a sphere of dimension at most $n^2+n-1$.