  In this paper we analyze the metastable behavior for the Ising model that
evolves under Kawasaki dynamics on the hexagonal lattice $\mathbb{H}^2$ in the
limit of vanishing temperature. Let $\Lambda\subset\mathbb{H}^2$ a finite set
which we assume to be arbitrarily large. Particles perform simple exclusion on
$\Lambda$, but when they occupy neighboring sites they feel a binding energy
$-U<0$. Along each bond touching the boundary of $\Lambda$ from the outside to
the inside, particles are created with rate $\rho=e^{-\Delta\beta}$, while
along each bond from the inside to the outside, particles are annihilated with
rate 1, where $\beta$ is the inverse temperature and $\Delta>0$ is an activity
parameter. For the choice $\Delta\in{(U,\frac{3}{2}U)}$ we prove that the empty
(resp.\ full) hexagon is the unique metastable (resp.\ stable) state. We
determine the asymptotic properties of the transition time from the metastable
to the stable state and we give a description of the critical configurations.
We show how not only their size but also their shape varies depending on the
thermodynamical parameters. Moreover, we emphasize the role that the specific
lattice plays in the analysis of the metastable Kawasaki dynamics by comparing
the different behavior of this system with the corresponding system on the
square lattice.
