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

Abstract

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 \(\tfrac12(k-1)(k-2)\). 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.