Energy-minimizing torus-valued maps with prescribed singularities, Plateau’s problem, and BV-lifting

This paper investigates the relationship between torus-valued maps with prescribed topological singularities and the mathematical formulations of Plateau’s problem, demonstrating that the infimum of the \(W^{1,1}\)-seminorm for maps with a given singularity equals the minimal mass of corresponding codimension-one normal currents with coefficients in \(\mathbb{R}^{k}\). Furthermore, it analyzes the BV-lifting problem, proving that the minimal \(BV\)-energy of liftings can be expressed through the minimal mass of codimension-one integral currents with coefficients in \(\mathbb{Z}^{k}\) bounded by the map's singularities.