Clifford Tori and the singularly perturbed Cahn-Hilliard equation

In this paper we construct entire solutions $u_{\varepsilon}$ to the Cahn-Hilliard equation $-\varepsilon^{2}Δ(-\varepsilon^{2}Δu+W^{'}(u))+W^{"}(u)(-\varepsilon^{2}Δu+W^{'}(u))=0$, under the volume constraint $\int_{\mathbb{R}^{3}}(1-u_{\varepsilon})dx=4\sqrt{2}π^{2}$, 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.

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