In this talk I will report on joint work with Sebastian Heller and Martin Traizet about recent progress in constructing higher genus harmonic maps using integrable systems methods. By taking the genus as parameter, we obtain a Taylor expansion for the area of the Lawson surfaces $(\xi_1,g)$ with an iterative algorithms to compute all coefficients. Remarkably, these coefficients turns out to be Multiple Zeta Values. Moreover, the Taylor series is shown to converge for genus $>2$ which also yields monotonicity of the area for $g>3$.