Classification of Minimal Immersions of Conformally Flat $3$-Tori and $4$-Tori in Spheres by The First Eigenfunctions

This paper is devoted to the study of minimal immersions of flat $n$-tori into spheres, especially those immersed by the first eigenfunctions (such immersion is called $\lambda_1$-minimal immersion), which also play important roles in spectral geometry. It is known that there are only two non-congruent $\lambda_1$-minimal $2$-tori in spheres, which are both flat. For higher dimensional case, the Clifford $n$-torus in $\mathbb{S}^{2n-1}$ might be the only known example in the literature. In this paper, by discussing the general construction of homogeneous minimal flat $n$-tori in spheres, we construct many new examples of $\lambda_1$-minimal flat $3$-tori and $4$-tori. In contrast to the rigidity in the case of $2$-tori, we show that there exists a $2$-parameter family of non-congruent $\lambda_1$-minimal flat $4$-tori. It turns out that the examples we constructed exhaust all $\lambda_1$-minimal immersions of conformally flat $3$-tori and $4$-tori in spheres. The classification involves some detailed investigations of shortest vectors in lattices, which can also be used to solve the Berger's problem on flat $3$-tori and $4$-tori. The dilation-invariant functional $\lambda_1(g)V(g)^{\frac{2}{n}}$ about the first eignvalue is proved to have maximal value among all flat $3$-tori and $4$-tori.