Communications in Analysis and Geometry

Volume 31 (2023)

Number 8

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

Pages: 1931 – 1968

DOI: https://dx.doi.org/10.4310/CAG.2023.v31.n8.a2

Authors

Rayssa Caju (Universidade Federal da Paraíba, João Pessoa, PB, Brazil; and Department of Mathematical Engineering, University of Chile, Santiago, Chile)

Pedro Gaspar (Department of Mathematics, University of Chicago, Illinois, U.S.A.; and Faculty of Mathematics, Pontificia Universidad Católica de Chile, Macul, Chile)

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).

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Received 18 October 2019

Accepted 2 September 2021

Published 10 August 2024