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arXiv · 2509.02739

On stable solutions to the Allen-Cahn equation with bounded energy density in $\mathbb{R}^4$

Enric Florit-Simon·Joaquim Serra
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Abstract

We show that stable solutions $u:\mathbb{R}^4\to (-1,1)$ 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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Enric Florit-Simon, Joaquim Serra. 2025-09-02. On stable solutions to the Allen-Cahn equation with bounded energy density in $\mathbb{R}^4$. https://arxiv.org/abs/2509.02739

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