# Harmonic maps and shift-invariant subspaces

## TL;DR

This paper explores the relationship between harmonic maps into the unitary group and their Grassmannian models, providing operator-theoretic criteria for finiteness of the uniton number with various applications.

## Contribution

It introduces a new operator-theoretic criterion for the finiteness of the uniton number in harmonic maps, linking harmonic analysis and geometric models.

## Key findings

- Derived a criterion for finiteness of the uniton number
- Connected harmonic maps with shift-invariant subspaces
- Presented applications of the criterion in geometric analysis

## Abstract

We investigate in detail the connection between harmonic maps from Riemann surfaces into the unitary group $\U(n)$ and their Grassmannian models: these are families of shift-invariant subspaces of $L^2(S^1,\C^n)$. With the help of operator-theoretic methods we derive a criterion for finiteness of the uniton number which has a large number of applications discussed in the paper.