Minimal surfaces and alternating multiple zetas

Abstract

In this paper we show for every sufficiently large integer $g$ the existence of a complete family of closed and embedded constant mean curvature (CMC) surfaces deforming the Lawson surfaces $\xi_{1,g}$ parametrized by their conformal type. When specializing to the minimal case, we discover a pattern resulting in the coefficients of the involved expansions being alternating multiple zeta values (MZVs), which generalizes the notion of Riemann's zeta values to multiple integer variables. This allows us to extend a new existence proof of the Lawson surfaces $\xi_{1,g}$ to all $g\geq 3$ using complex analytic methods and to give closed form expressions of their area expansion up to order $7$. For example, the third order coefficient is $\tfrac{9}{4}\zeta(3)$ (the first and second order term were shown to be $\log(2)$ and $0$ respectively in \cite{HHT}). As a corollary, we obtain that the area of $\xi_{1,g}$ is monotonically increasing in their genus $g$ for all $g\geq 0.$