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

Abstract

We make use of the flexibility of infinite-index solutions to the Allen-Cahn equation 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^d whose zero level set has a connected component diffeomorphic (and arbitrarily close) to a rescaling of $\Sigma$. More generally, we prove the existence of solutions with a finite number of compact connected components of prescribed topology in their zero level sets.