Clifford Tori and the singularly perturbed Cahn-Hilliard equation

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In this paper we construct entire solutions uε to the Cahn-Hilliard equation -ε²Δ(-ε²Δu+W'(u))+W"(u)(-ε²Δu+W'(u))=0, under the volume constraint ∫ᵣ³(1-uε)dx=4√2π², whose nodal set approaches the Clifford Torus, that is the Torus with radii of ratio 1/√2 embedded in R³, as ε→ 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.