Corpus ID: 189898181

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

@article{Caju2019SolutionsOT,
  title={Solutions of the Allen-Cahn equation on closed manifolds in the presence of symmetry},
  author={Rayssa Caju and P. Gaspar},
  journal={arXiv: Differential Geometry},
  year={2019}
}
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-Ritore (in the case of closed manifolds and zero mean curvature). 
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