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We consider the Glauber dynamics for the Ising model with "+" boundaryconditions, at zero temperature or at temperature which goes to zero with thesystem size (hence the quotation marks in the title). In dimension d=3 we provethat an initial domain of linear size L of "-" spins disappears within a time\tau_+ which is at most L^2(\log L)^c and at least L^2/(c\log L), for some c>0.The proof of the upper bound proceeds via comparison with an auxiliary dynamicswhich mimics the motion by mean curvature that is expected to describe, onlarge time-scales, the evolution of the interface between "+" and "-" domains.The analysis of the auxiliary dynamics requires recent results on thefluctuations of the height function associated to dimer coverings of theinfinite honeycomb lattice. Our result, apart from the spurious logarithmicfactor, is the first rigorous confirmation of the expected behavior\tau_+\simeq const\times L^2, conjectured on heuristic grounds. In dimensiond=2, \tau_+ can be shown to be of order L^2 without logarithmic corrections:the upper bound was proven in [Fontes, Schonmann, Sidoravicius, 2002] and herewe provide the lower bound. For d=2, we also prove that the spectral gap of thegenerator behaves like c/L for L large, as conjectured in [Bodineau-Martinelli,2002].

"Zero" temperature stochastic 3D Ising model and dimer covering fluctuations: a first step towards interface mean curvature motion