  Let $\DD$ be a simply connected, smooth enough domain of $\bbR^2$. For $L>0$
consider the continuous time, zero-temperature heat bath dynamics for the
nearest-neighbor Ising model on $\mathbb Z^2$ with initial condition such that
$\sigma_x=-1$ if $x\in L\DD$ and $\sigma_x=+1$ otherwise. It is conjectured
\cite{cf:Spohn} that, in the diffusive limit where space is rescaled by $L$,
time by $L^2$ and $L\to\infty$, the boundary of the droplet of "$-$" spins
follows a \emph{deterministic} anisotropic curve-shortening flow, where the
normal velocity at a point of its boundary is given by the local curvature
times an explicit function of the local slope. The behavior should be similar
at finite temperature $T<T_c$, with a different temperature-dependent
anisotropy function. We prove this conjecture (at zero temperature) when $\DD$
is convex. Existence and regularity of the solution of the deterministic
curve-shortening flow is not obvious \textit{a priori} and is part of our
result. To our knowledge, this is the first proof of mean curvature-type
droplet shrinking for a model with genuine microscopic dynamics.
