Clifford Tori and the singularly perturbed Cahn-Hilliard equation

In this paper we construct entire solutions $u_{\varepsilon}$ to the Cahn-Hilliard equation $-\varepsilon^{2}\Delta(-\varepsilon^{2}\Delta u+W^{'}(u))+W^{"}(u)(-\varepsilon^{2}\Delta u+W^{'}(u))=0$, under the volume constraint $\int_{\mathbb{R}^{3}}(1-u_{\varepsilon})dx=4\sqrt{2}\pi^{2}$, whose nodal set approaches the Clifford Torus, that is the Torus with radii of ratio $1/\sqrt{2}$ embedded in $\mathbb{R}^{3}$, as $\varepsilon\to 0$. What is crucial is that the Clifford Torus is a Willmore hypersurface and it is non-degenerate, up to conformal transformations. The proof is based on the Lyapunov-Schmidt reduction and on careful geometric expansions of the laplacian.