Non-algebraicity of the Lawson surfaces ${\xi }_{k,l}$

Abstract

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

arXiv comment24 pages. Comments are welcome