Minimal surfaces and alternating multiple zetas

In this talk we give an alternate existence proof of Lawson surfaces ${\xi }_{1,g}$ for $g\ge 3$ using complex analytic methods. When computing the Taylor approximation at $g=\mathrm{\infty }$ we discover a pattern resulting in the coefficients of the involved expansions being alternating multiple zeta values (MZVs), a generalised notion of Riemann's zeta values to multiple integer variables. When specialising to the Taylor expansion of the area, we find for example that the third order coefficient is $\frac{9}{4}\zeta (4)$ (the first and second order tern are log(2) and 0, respectively). As a corollary of these higher order expansions, we obtain that the area of ${\xi }_{1,g}$ is monotonically increasing in their genus for all $g\ge 0$. This is joint work with Steven Charlton, Sebastian Heller and Martin Traizet.