The planar surface tension problem for the Gates-Penrose-Lebowitz free energy function concerns the minimization of this functional for profiles $m(x,y)$ on a cylinder in $R\times C\in R^d$ with cubic cross section $C$ and periodic boundary conditions. It has been shown by Alberti and Belletini that the only minimizing profiles are of the form $m(x,y) = n(x)$ where $x \in R$ and $y \in C$ and $n$ is the instanton for the one dimensional GPL functional. As far as dynamical stability of the minimizers is concerned, the case of Glauber dynamics (spin flips) is by now well understood. However, the case of Kawasaki dynamics (spin exchanges) is different, in particular because of the conservation law and the lack of a spectral gap. We present a proof of dynamical stability in this case that is joint work with Enza Orlandi.