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

In this paper, we investigate the relation between energy-minimizing torus-valued maps with prescribed singularities, the lifting problem for torus-valued maps in the space BV, and Plateau's problem for vectorial currents, in codimension one. First, we show that the infimum of the $W^{1,1}$-seminorm among all maps with values in the $k$-dimensional flat torus and prescribed topological singularities $S$ is equal to the minimum of the mass among all $\textit{normal}$ $\mathbb{R}^k$-currents, of codimension one, bounded by $S$. Then, we show that the minimum of the $BV$-energy among all liftings of a given torus-valued $W^{1,1}$-map $\textbf{u}$ can be expressed in terms of the minimum mass among all $\textit{integral}$ $\mathbb{Z}^k$-currents, of codimension one, bounded by the singularities of $\textbf{u}$. As a byproduct of our analysis, we provide a bound for the solution of the integral Plateau problem, in codimension one, in terms of Plateau's problem for normal currents.