Solutions of the Allen-Cahn equation on closed manifolds in the presence of symmetry (Q6628905)

The main result of this paper is, given a minimal hypersurface \(\Gamma\) in a closed Riemannian manifold \((M,g)\), if \(\Gamma\) separates \(M\) and all Jacobi fields on \(\Gamma\) are generated by global isometries, then there exists a sequence of solutions \(u_\varepsilon\) to the singularly perturbed Allen-Cahn equation on \(M\), which converges to \(\Gamma\) in a suitable sense. A relation between the Morse index of \(u_\varepsilon\) and \(\Gamma\) is also established. This generalizes the result of \textit{F. Pacard} and \textit{M. Ritoré} [J. Differ. Geom. 64, No. 3, 359--423 (2003; Zbl 1070.58014)], where the nondegeneracy condition for \(\Gamma\) (i.e., non-existence of Jacobi fields on \(\Gamma\)) was needed.