The symmetric space X_n= SL(n,)/ SO(n) can be interpreted as the Teichmüller space of marked, unit volume, flat n-dimensional tori. It comes with a unique (up to scale) SL(n,)-invariant metric dXn. In 1939 Teichmüller gave a modular interpretation of dX2
Nearby in the stack
(the hyperbolic metric) in terms of an extremal mapping problem for quasiconformal dilatation. Such a modular interpretation for
dXn
for
n≥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 total expansion
(f)∈ [0,∞]
of a Lipschitz map
f:M→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:₀→₁
between
n
-dimensional, flat, unit-volume tori, we prove that
(f)
is minimized in the homotopy class of
f
precisely by the affine maps in that class and takes on these the value
dXn
. We prove similar results for the Hilbert-Schmidt expansion