Embedded special Legendrian surfaces in ${\mathbb{S}}^5$

We construct the first smooth embedded compact special Legendrian surfaces in ${\mathbb{S}}^5$ of genus greater than one. More precisely, for every sufficiently large integer $k$, we construct an embedded special Legendrian surface whose conformal structure is the Fermat curve of degree $k$ and genus $\frac{1}{2}(k-1)(k-2)$. Our approach combines an elementary implicit function theorem with the description of special Legendrian surfaces via loop algebra-valued meromorphic connections and a characterization of the unitarizability locus in the ${SL}_3(\mathbb{C})$-character variety of the thrice-punctured sphere.