Classification of minimal immersions of conformally flat 3-tori and 4-tori into spheres by the first eigenfunctions

The Clifford 2-torus in S³ and the equlaterial 2-torus in S⁵ are known as the only minimal immersions of 2-tori into spheres by the first eigenfunctions (called λ₁-minimal for short). For n≥3, the Clifford n-torus in S^2n-1 might be the only known example of λ₁-minimal n-tori in the literature. By discussing the general construction of homogeneous minimal flat n-tori in spheres, we construct several new examples of λ₁-minimal flat 3-tori and 4-tori. In particular, the existence of 2-parameter family of non-congruent λ₁-minimal flat 4-tori is shown for the first time. We obtain the complete classification for λ₁-minimal immersions of conformally flat 3-tori and 4-tori in spheres, by some detailed investigations of shortest vectors in lattices, which could be of independent interests. Using them, we also solve the Berger’s problem (finding the maximal value of the dilation-invariant functional λ₁(g)V(g)²ⁿ) among all flat 3-tori and 4-tori.