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We prove the existence of multiple solutions to the Allen--Cahn--Hilliard(ACH) vectorial equation (with two equations) involving a triple-well(triphasic) potential with a small volume constraint on a closed parallelizableRiemannian manifold. More precisely, we find a lower bound for the number ofsolutions depending on some topological invariants of the underlying manifold.The phase transition potential is considered to have a finite set of globalminima, where it also vanishes, and a subcritical growth at infinity. Ourstrategy is to employ the Lusternik--Schnirelmann and infinite-dimensionalMorse theories for the vectorial energy functional. To this end, we exploitthat the associated ACH energy $\Gamma$-converges to the weightedmulti-perimeter for clusters, which combined with some deep theorems fromisoperimetric theory yields the suitable setup to apply the photography method.Along the way, the lack of a closed analytic expression for themulti-isoperimetric function for clusters imposes a delicate issue.Furthermore, using a transversality theorem, we also show the genericity of theset of metrics for which solutions to the ACH system are nondegenerate.

Multiplicity of solutions to the multiphasic Allen-Cahn-Hilliard system with a small volume constraint on closed parallelizable manifolds