A duality for minimal surfaces in the Heisenberg group
Shimpei Kobayashi
TL;DR
The paper develops a duality theory for nowhere-vertical minimal surfaces in the Heisenberg group Nil_3 by introducing dual generating spinors and a corresponding dual surface f^*. It shows that the dual surface is again minimal and nowhere-vertical, with f^{**}=f up to rigid motion, and provides explicit data relations such as $B^*=B$, $g^*=g$, $h^*={16|B|}/{h}$, and $e^{u^*}=4^4|B|^2/h^4\, e^u$. Using the loop group formalism, the authors derive a dual Sym formula from the spectral-parameter derivative of the extended frame, yielding a family of minimal surfaces $f_{\pm}^{\lambda}$ in Nil_3 with $f_{-}^{\lambda}|_{\lambda=1}=f$ and $f_{+}^{\lambda}|_{\lambda=1}=f^*$, and they illustrate the theory with self-dual examples (hyperbolic paraboloid, helicoid) and Smyth-type surfaces. This work connects minimal surface theory in Nil_3 with harmonic maps into $\mathbb{H}^2$ and integrable-systems methods, enriching the notion of duality in non-Euclidean geometries. The results provide a systematic construction of dual minimal surfaces in Nil_3 and a concrete Sym-formula framework for their analysis and examples.
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.