Minimal tori in five-dimensional sphere in $C^3$

TL;DR

This paper studies a special class of minimal tori in a 5D sphere within complex 3-space, deriving integrable equations and constructing finite-gap solutions using inverse scattering.

Contribution

It reduces the immersion equations for these minimal tori to an integrable PDE and constructs explicit finite-gap solutions, advancing understanding of minimal surfaces in higher dimensions.

Findings

Immersion equations reduce to an integrable PDE

Finite-gap minimal tori are explicitly constructed

The inverse scattering method is applied successfully

Abstract

Special class of surfaces in five-dimensional sphere in $C^3$ is considered. Immersion equations for minimal tori of that class are shown to be reducible to the equation $u_{z\bar z}=e^u-e^{-2u}$ which is integrable by means of inverse scattering method. Finite-gap minimal tori are constructed.