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In this paper, we investigate the relation between energy-minimizingtorus-valued maps with prescribed singularities, the lifting problem fortorus-valued maps in the space BV, and Plateau's problem for vectorialcurrents, 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 torusand prescribed topological singularities $S$ is equal to the minimum of themass 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 allliftings of a given torus-valued $W^{1,1}$-map $\textbf{u}$ can be expressed interms 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 thesolution of the integral Plateau problem, in codimension one, in terms ofPlateau's problem for normal currents.

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