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

In this paper we establish an improvement of the 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,α regularity of flat level sets. This generalizes Allard's ε-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 ℝn for n ≤ 7 is also given. [ABSTRACT FROM AUTHOR]