Integral iterations for harmonic maps

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.