Shows stable solutions to the Allen-Cahn equation are one-dimensional, implying geometric insights into phase transitions.
We show that stable solutions u:R⁴→ (-1,1) to the Allen-Cahn equation with bounded energy density (or equivalently, with cubic energy growth) are one-dimensional. This is known to entail important geometric consequences, such as robust curvature estimates for stable phase transitions, and the multiplicity one and Morse index conjectures of Marques-Neves for Allen-Cahn approximations of minimal hypersurfaces in closed 4-manifolds.
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Florit-Simon et al. (2025) studied this question.
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