Mathematical analysis demonstrates rigidity of isotropic harmonic maps on complex projective spaces, indicating structural stability for towers of harmonic bands in condensed matter physics.
We prove the rigidity of isotropic harmonic maps from a 2-torus to a complex projective space, when they are constructed from holomorphic embeddings associated to complete linear systems. We also prove that this rigidity holds for any holomorphic embeddings without special hyperosculation points, with an extra assumption on the pullbacks of Fubini--Study symplectic forms. These results ensure the rigidity of towers of harmonic bands in condensed matter physics.
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Hashimoto et al. (2026) studied this question.