Generating Axially Symmetric Minimal Hyper-surfaces in R^{1,3}

TL;DR

This paper demonstrates a method to generate infinitely many axially symmetric minimal hypersurfaces in 4D Minkowski space using symmetry transformations analogous to classical Baecklund transformations, revealing new geometric insights.

Contribution

It introduces a novel approach combining scaling symmetries and a specific involutive transformation to produce minimal hypersurfaces from any given example.

Findings

Infinite family of axially symmetric minimal hypersurfaces generated

Symmetry transformations analogous to Baecklund transformations identified

New geometric properties of minimal hypersurfaces in Minkowski space uncovered

Abstract

It is shown that, somewhat similar to the case of classical Baecklund transformations for surfaces of constant negative curvature, infinitely many axially symmetric minimal hypersurfaces in 4-dimensional Minkowski-space can be obtained, in a non-trivial way, from any given one by combining the scaling symmetries of the equations in light cone coordinates with a non-obvious symmetry (the analogue of Bianchis original transformation) - which can be shown to be involutive/correspond to a space-reflection.