Non-algebraicity of the Lawson surfaces $ξ_{k,l}$

Abstract

We prove that every Lawson surface $ξ_{k,l}\subset\mathbb S^3$ of genus $kl>1$ is non-algebraic: any real polynomial vanishing on $ξ_{k,l}$ vanishes identically on $\mathbb S^3$. This answers a question posed by Hsiang in 1967 and proves Lawson's 1970 conjecture for $ξ_{2,2}$. The proof uses a reflection principle for the monodromy of the associated family of flat connections.