Extremal mappings of tori, Teichmüller potentials and symmetric-space distance

The symmetric space $X_n=\mathrm{S}\mathrm{L}(n,\text{Rb})\mathrm{/}\mathrm{S}\mathrm{O}(n)$ can be interpreted as the Teichmüller space of marked, unit volume, flat $n$-dimensional tori. It comes with a unique (up to scale) $\mathrm{S}\mathrm{L}(n,\text{Rb})$-invariant metric $d_{X_n}$. In 1939 Teichmüller gave a modular interpretation of $d_{X_2}$ (the hyperbolic metric) in terms of an extremal mapping problem for quasiconformal dilatation. Such a modular interpretation for $d_{X_n}$ for $n\ge 3$ has remained unaddressed: the natural candidates - minimal quasiconformal dilatation, Lipschitz constant, or total energy - do not work. In this paper we give such a modular interpretation, two in fact. We introduce the {\em total expansion} $\text{TE}(f)\in [0,\mathrm{\infty }]$ of a Lipschitz map $f:M\to N$ between Riemannian manifolds, a notion related to the notion of ``$k$-dilatation’’ developed by Gromov, Guth and others. For volume-preserving Lipschitz maps $f:{\text{Tc}}_0\to {\text{Tc}}_1$ between $n$-dimensional, flat, unit-volume tori, we prove that $\text{TE}(f)$ is minimized in the homotopy class of $f$ precisely by the affine maps in that class and takes on these the value $d_{X_n}$. We prove similar results for the \emph{Hilbert-Schmidt expansion} $\text{HE}(f)$, which is a simple integral over $M$ and has more of an $L^2$ flavor.