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

TL;DR

This paper constructs the first smooth embedded compact special Legendrian surfaces in  with genus greater than one, using advanced geometric and algebraic techniques.

Contribution

It introduces a novel construction method for high-genus embedded special Legendrian surfaces in , combining implicit function theorem and loop algebra techniques.

Findings

Constructed embedded special Legendrian surfaces of genus > 1 in .

Used loop algebra-valued meromorphic connections for characterization.

Identified unitarizability locus in SL_3()-character variety.

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.