Solutions of the Allen-Cahn equation on closed manifolds in the presence of symmetry

Abstract

We prove that given a minimal hypersurface $\Gamma$ in a compact Riemannian manifold $M$ without boundary, if all the Jacobi fields of $\Gamma$ are generated by ambient isometries, then we can find solutions of the Allen-Cahn equation $-\varepsilon^2\Delta u +W'(u)=0$ on $M$, for sufficiently small $\varepsilon>0$, whose nodal sets converge to $\Gamma$. This extends the results of Pacard-Ritor\'e (in the case of closed manifolds and zero mean curvature).