A duality for minimal surfaces in the Heisenberg group

2025/04/29 by Shimpei Kobayashi, Kobayashi, Shimpei
Mathematics · #53A10 #53C42 #58D10 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2504.20671

openalex publication_date 2025/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce and study the notion of a transformation surface associated with a nowhere-vertical minimal surface in the three-dimensional Heisenberg group, and prove its minimality and duality. Furthermore, by using the logarithmic derivative of the moving frame with respect to the spectral parameter, we derive the Sym formula for the dual minimal surface.

Citations

• A loop group method for minimal surfaces in the three-dimensional Heisenberg group
• Holomorphic Representation of Constant Mean Curvature Surfaces in Minkowski Space: Consequences of Non-Compactness in Loop Group Methods
• Surfaces in three-dimensional Lie groups
• Bonnet pairs and isothermic surfaces

Related

• Minimal surfaces with non-trivial topology in the three-dimensional Heisenberg group
• Ruled minimal surfaces in the three dimensional Heisenberg group
• The Gauss map of Minimal graphs in the Heisenberg group
• Minimal null scrolls in the three-dimensional Heisenberg group
• Timelike minimal surfaces in the three-dimensional Heisenberg group