  We consider a general class of Glauber dynamics reversible with respect to
the standard Ising model in $\bbZ^d$ with zero external field and inverse
temperature $\gb$ strictly larger than the critical value $\gb_c$ in dimension
2 or the so called ``slab threshold'' $\hat \b_c$ in dimension $d \geq 3$. We
first prove that the inverse spectral gap in a large cube of side $N$ with plus
boundary conditions is, apart from logarithmic corrections, larger than $N$ in
$d=2$ while the logarithmic Sobolev constant is instead larger than $N^2$ in
any dimension. Such a result substantially improves over all the previous
existing bounds and agrees with a similar computations obtained in the
framework of a one dimensional toy model based on mean curvature motion. The
proof, based on a suggestion made by H.T. Yau some years ago, explicitly
constructs a subtle test function which forces a large droplet of the minus
phase inside the plus phase. The relevant bounds for general $d\ge 2$ are then
obtained via a careful use of the recent $\bbL^1$--approach to the Wulff
construction. Finally we prove that in $d=2$ the probability that two
independent initial configurations, distributed according to the infinite
volume plus phase and evolving under any coupling, agree at the origin at time
$t$ is bounded from below by a stretched exponential $\exp(-\sqrt{t})$, again
apart from logarithmic corrections. Such a result should be considered as a
first step toward a rigorous proof that, as conjectured by Fisher and Huse some
years ago, the equilibrium time auto-correlation of the spin at the origin
decays as a stretched exponential in $d=2$.
