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 without boundary, if all the Jacobi fields of $\Gamma$ are generated by ambient isometries, then we can find solutions of the Allen–Cahn equation $-\varepsilon^2 \Delta u + W^\prime (u) = 0$ on $M$, for sufficiently small $\varepsilon \gt 0$, whose nodal sets converge to $\Gamma$. This extends the results of Pacard–Ritoré $\href{https://doi.org/10.4310/jdg/1090426999}{[41]}$ (in the case of closed manifolds and zero mean curvature).