English

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

Differential Geometry 2026-04-24 v1 Mathematical Physics Algebraic Geometry math.MP

Abstract

We construct the first smooth embedded compact special Legendrian surfaces in S5\mathbb S^5 of genus greater than one. More precisely, for every sufficiently large integer kk, we construct an embedded special Legendrian surface whose conformal structure is the Fermat curve of degree kk and genus 12(k1)(k2)\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 SL3(C){SL}_{3}(\mathbb C)-character variety of the thrice-punctured sphere.

Keywords

Cite

@article{arxiv.2604.21521,
  title  = {Embedded special Legendrian surfaces in $\mathbb S^5$},
  author = {Sebastian Heller and Franz Pedit and Charles Ouyang},
  journal= {arXiv preprint arXiv:2604.21521},
  year   = {2026}
}

Comments

64 pages; comments welcome

R2 v1 2026-07-01T12:32:14.532Z