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

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The Clifford 2-torus in \(\mathbb {S}^3\) and the equlaterial 2-torus in \(\mathbb {S}^5\) are known as the only minimal immersions of 2-tori into spheres by the first eigenfunctions (called \(\lambda _1\)-minimal for short). For \(n\ge 3\), the Clifford n-torus in \(\mathbb {S}^{2n-1}\) might be the only known example of \(\lambda _1\)-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 \(\lambda _1\)-minimal flat 3-tori and 4-tori. In particular, the existence of 2-parameter family of non-congruent \(\lambda _1\)-minimal flat 4-tori is shown for the first time. We obtain the complete classification for \(\lambda _1\)-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 \(\lambda _1(g)V(g)^{\frac{2}{n}}\)) among all flat 3-tori and 4-tori.

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Acknowledgements

The first author and the third author is supported by NSFC No. 12171473. The second author is supported by NSFC No. 12371052 and No. 11971107. The authors are grateful to Prof. C.P. Wang for his continuous encouragement on this work. The authors are thankful to Prof. Q.-S. Chi and Prof. R. Kusner for valuable discussions. We appreciate the referee for many valuable suggestions.

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Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, People’s Republic of China

    Ying Lü

  2. School of Mathematics and Statistics, FJKLMAA, Key Laboratory of Analytical Mathematics and Applications (Ministry of Education), Fujian Normal University, Fuzhou, 350117, People’s Republic of China

    Peng Wang

  3. School of Mathematical Sciences, Beihang University, Beijing, 100191, People’s Republic of China

    Zhenxiao Xie

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  1. Ying Lü
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Correspondence to Zhenxiao Xie.

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Lü, Y., Wang, P. & Xie, Z. Classification of minimal immersions of conformally flat 3-tori and 4-tori into spheres by the first eigenfunctions. Math. Ann. (2024). https://doi.org/10.1007/s00208-024-02799-8

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