That's cool but by that argument can't i just fold a Mercator map in half and also have no boundary cut?
The problem statement is: find a mapping from the surface of a sphere to ℝ² that minimizes a particular penalty function. This paper maps each hemisphere to ℝ², and then argues that the normal boundary penalty term can be ignored.
However, if you just look at what the map does to South America and Africa, where there's a massive discontinuity at the equator, it's absurd to argue that the boundary penalty should be ignored. This map is useless for equatorial regions, and the penalty function should reflect that.
The problem that this paper is trying to solve, however, is the creation of a minimally distorted (according to some penalty function) global map.
As far as I can tell its not published anywhere nor received any peer review.
You need to both fold it in half and glue the ends together, basically creating a torus (or two-sided cylinder) shape