  We study the Glauber dynamics for the zero-temperature Ising model in
dimension d=4 with "plus" boundary condition.Let T+ be the time needed for an
hypercube of size L entirely filled with "minus" spins to become entirely
"plus". We prove that T+ is O(L^2(log L)^c) for some constant c, not depending
on the dimension. This brings further rigorous justification for the so-called
"Lifshitz law" T+ = O(L^2) [5, 3] conjectured on heuristic grounds. The key
point of our proof is to use the detail knowledge that we have on the
three-dimensional problem: results for fluctuation of monotone interfaces at
equilibrium and mixing time for monotone interfaces dynamics extracted from
[2], to get the result in higher dimension.
