An ∞-Laplacian for differential forms, and calibrated laminations

Motivated by Thurston and Daskalopoulos--Uhlenbeck's approach to Teichm\"uller theory, we study the behavior of q-harmonic functions and their p-harmonic conjugates in the limit as q → 1, where $1/p + 1/q = 1$. The $1$-Laplacian is already known to give rise to laminations by minimal hypersurfaces; we show that the limiting p-harmonic conjugates converge to calibrations F of the laminations. Moreover, we show that the laminations which are calibrated by F are exactly those which arise from the $1$-Laplacian. We also explore the limiting dual problem as a model problem for the optimal Lipschitz extension problem, which exhibits behavior rather unlike the scalar ∞-Laplacian. In a companion work, we will apply the main result of this paper to associate to each class in Hd - 1 a lamination in a canonical way, and study the duality of the stable norm on Hd - 1.