# Integral iterations for harmonic maps

## TL;DR

This paper investigates minimal harmonic maps from the complex plane into a symmetric space, linking their asymptotic geometry to convex polygons via integral fixed-point problems and spectral network techniques.

## Contribution

It introduces a conjectural method to determine asymptotic polygons of harmonic maps using integral operators and spectral network technology.

## Key findings

- Explicit examples of asymptotic polygons are computed.
- A fixed-point approach for asymptotic structure is proposed.
- Spectral networks are utilized to formulate the problem.

## Abstract

We study minimal harmonic maps $g: {\mathbb{C}} \to SO(3) \backslash SL(3,{\mathbb{R}})$, parameterized by polynomial cubic differentials $P$ in the plane. The asymptotic structure of such a $g$ is determined by a convex polygon $Y(P)$ in ${\mathbb{RP}^2}$. We give a conjectural method for determining $Y(P)$ by solving a fixed-point problem for a certain integral operator. The technology of spectral networks and BPS state counts is a key input to the formulation of this fixed-point problem. We work out two families of examples in detail.