Explicit construction of harmonic two-spheres into the complex Grassmannian

TL;DR

This paper provides an explicit algebraic method to construct all harmonic maps of finite uniton number from a Riemann surface into complex Grassmannians, based on meromorphic functions and derivatives.

Contribution

It introduces a new explicit construction technique for harmonic maps into complex Grassmannians, including a sharp estimate of their uniton number.

Findings

All harmonic maps can be constructed from meromorphic functions and derivatives.

A sharp estimate for the uniton number depending on initial data.

Explicit algebraic formulas for harmonic maps into Grassmannians.

Abstract

We present an explicit description of all harmonic maps of finite uniton number from a Riemann surface into a complex Grassmannian. Namely, starting from a constant map $Q$ and a collection of meromorphic functions and their derivatives, we show how to algebraically construct all harmonic maps from the two-sphere into a given Grassmannian $G_p(\mathbb C^n)$. In this setting the uniton number depends on $Q$ and $p$ and we obtain a sharp estimate for it.