Harmonic band theory: rigidity of non-zero degree harmonic maps from 2-torus to complex projective space

TL;DR

This paper proves the rigidity of certain harmonic maps from a 2-torus to complex projective space, with implications for condensed matter physics, by analyzing holomorphic embeddings and symplectic form pullbacks.

Contribution

It establishes new rigidity results for harmonic maps constructed from holomorphic embeddings, extending previous understanding in geometric analysis.

Findings

Rigidity of isotropic harmonic maps from 2-torus to complex projective space.

Rigidity holds for holomorphic embeddings without hyperosculation points under certain conditions.

Ensures rigidity of harmonic bands in condensed matter physics.

Abstract

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.