Uniform stability in the Euclidean isoperimetric problem for the Allen–Cahn energy

We consider the isoperimetric problem defined on the whole ${ℝ}^n$ by the Allen–Cahn energy functional. For nondegenerate double-well potentials, we prove sharp quantitative stability inequalities of quadratic type which are uniform in the length scale of the phase transitions. We also derive a rigidity theorem for critical points analogous to the classical Alexandrov theorem for constant mean curvature boundaries.