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We make use of the flexibility of infinite-index solutions to the Allen-Cahnequation to show that, given any compact hypersurface $\Sigma$ of R^d, with$d\geq 4$, there is a bounded entire solution of the Allen-Cahn equation on R^dwhose zero level set has a connected component diffeomorphic (and arbitrarilyclose) to a rescaling of $\Sigma$. More generally, we prove the existence ofsolutions with a finite number of compact connected components of prescribedtopology in their zero level sets.

Bounded solutions to the Allen-Cahn equation with level sets of any compact topology