  Let M be any closed, locally symmetric n-manifold (n>1) of nonpositive
curvature. Assume that M has no locally Euclidean factors and no factors
locally isometric to SL(3,R). Then for any closed Riemannian manifold N and any
continuous map f:N -> M, we show there is a C^1 representative in the homotopy
class of f with Jacobian bounded by a universal constant C depending only on n
and the smallest Ricci curvatures of N and M. This implies that deg(f)<= C
Vol(N)/Vol(M). For M negatively curved this was proved by Gromov. Two
corollaries of our result are that Minvol(N)>0 whenever deg(f)<>0 and a simple
proof of G. Prasad's result that lattices in semi-simple Lie groups are
co-Hopfian. We prove a version of all results for finite volume manifolds as
well.
