Survey on topological methods for Allen--Cahn equations and systems

Abstract

We present a survey on multiplicity results for the Allen--Cahn equation and systems in the singular perturbation regime, emphasizing their geometric interpretation through $\Gamma$-convergence and isoperimetric theory. In the scalar case, the Allen--Cahn functional converges to perimeter, giving rise to minimal and constant-mean-curvature hypersurfaces, while vectorial Allen--Cahn systems lead to multi-phase isoperimetric clusters. The main methodological tool discussed is the photography method, a variational-topological approach based on localized approximate solutions and barycenter maps, which enables one to encode the topology of the ambient manifold into multiplicity results. We compare problems posed on closed manifolds with those on manifolds with boundary, describing the distinct geometric effects induced by Neumann and Dirichlet boundary conditions. The survey highlights both the effectiveness and the limitations of this framework, particularly in the vectorial case, where the lack of a full classification of isoperimetric clusters creates fundamental analytical challenges.