Clifford Tori and the singularly perturbed Cahn–Hilliard equation

In this paper we construct entire solutions us to the Cahn-Hilliard equation $-epsilon(2)Delta(-epsilon(2)Delta u + W '(u)) + W ''(u)(epsilon(2)Delta u + W '(u)) = epsilon(4)lambda(epsilon) (1 - u(epsilon))$, under the volume constraint $integral(R3) (1 - u(epsilon))(2) dx = 8 root 2 pi(2)c(epsilon)$, with $c(epsilon) -> 1$ as $epsilon -> 0$, whose nodal set approaches the Clifford Torus, that is the Torus with radii of ratio $1/root 2$ embedded in $R-3$, as $epsilon -> 0$. It is crucial 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.