A general comparison theorem for $p$-harmonic maps in homotopy class

We prove a general comparison result for homotopic finite $p$-energy $C^1$ $p$-harmonic maps $u,v:M\to N$ between Riemannian manifolds, assuming that $M$ is $p$-parabolic and $N$ is complete and non-positively curved. In particular, we construct a homotopy through constant $p$-energy maps, which turn out to be $p$-harmonic when $N$ is compact. Moreover, we obtain uniqueness in the case of negatively curved $N$. This generalizes a well known result in the harmonic setting due to R. Schoen and S.T. Yau.