A new proof of Savin's theorem on Allen–Cahn equations

Source abstract

In this paper we establish an improvement of tilt-excess decay estimate for the Allen–Cahn equation, and use this to give a new proof of Savin's theorem on the uniform C^{1,\alpha} regularity of flat level sets. This generalizes Allard’s \varepsilon -regularity theorem for stationary varifolds to the setting of Allen–Cahn equations. A new proof of Savin’s theorem on the one-dimensional symmetry of minimizers in \mathbb{R}^n for n\leq 7 is also given.