Stable De Giorgi conjecture of the Allen--Cahn equation in $\mathbb{R}^3$

We prove that every bounded stable entire solution $v$ of the Allen--Cahn equation in $\R^3$ is one-dimensional. As a consequence, the full De Giorgi conjecture in $\mathbb{R}^4$ is true. We also obtain local curvature estimates for stable solutions. The proof strategy is inspired by the recent brea