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

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