Title: Periodic solutions to the Cahn-Hilliard equation in the plane,
Abstract: In this paper we construct entire solutions to the Cahn-Hilliard equation
$-\Delta(-\Delta u+W^{'}(u))+W^{"}(u)(-\Delta u+W^{'}(u))=0$ in the Euclidean
plane, where $W(u)$ is the standard double-well potential $\frac{1}{4}
(1-u^2)^2$. Such solutions have a non-trivial profile that shadows a Willmore
planar curve, and converge uniformly to $\pm 1$ as $x_2 \to \pm \infty$. These
solutions give a counterexample to the counterpart of Gibbons' conjecture for
the fourth-order counterpart of the Allen-Cahn equation. We also study the
$x_2$-derivative of these solutions using the special structure of Willmore's
equation.