2002/11/27 by Sung Ho Wang, Wang, Sung Ho
Mathematics · Physics and Astronomy · #53C25 #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.AP #math.DG #msc:53C25
paper · pdf · doi:10.48550/arxiv.math/0211439
42 pages
openalex publication_date 2002/11/27 · arxiv created 2003/05/04 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A surface Σ⊂ S5 ⊂ ℂ3 is called special Legendrian if the cone 0 × Σ⊂ ℂ3 is special Lagrangian. The purpose of this paper is to propose a general method toward constructing compact special Legendrian surfaces of high genus. It is proved \emphthere exists a compact, orientable, Hamiltonian stationary Lagrangian surface of genus 1+(k(k-3))/(2) in ℂP2 for each integer k ≥ 3, which is a smooth branched surface except at most finitely many conical singularities. If this surface is smooth, it is minimal and the Legendrian lift of the surface is the desired compact special Legendrian surface. We first establish the existence of a minimizer of area among Lagrangian disks in a relative homotopy class of a Kähler-Einstein surface without Lagrangian homotopy classes with respect to a configuration Γ that consists of the fixed point loci of Kähler involutions. Γ in addition must satisfy certain null relative homotopy conditions and angle criteria. The fundamental domain thus obtained is smooth along the boundary, and has finitely many interior singular points. We then apply successive reflection of this fundamental domain along its boundary to obtain a complete or compact Lagrangian surface.