We consider Cauchy data
(g,)
on
3
that are asymptotically Euclidean and that satisfy the vacuum constraint equations of general relativity. Only those
(g,)
are treated that can be joined by a curve of "sufficiently bounded" initial data to the trivial data
(, 0)
. It is shown that in the Cauchy developments of such data, the maximal slicing condition
tr = 0
can always be satisfied. The proof uses the recently introduced "weighted Sobolev spaces" of Nirenberg, Walker, and Cantor.