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

TL;DR

This paper establishes sharp quantitative stability inequalities for the Allen--Cahn energy in the Euclidean isoperimetric problem, demonstrating uniformity across phase transition scales and proving a rigidity theorem for critical points.

Contribution

It introduces uniform stability inequalities for the Allen--Cahn energy and a rigidity theorem for critical points, extending classical geometric results.

Findings

Proved sharp quadratic stability inequalities for Allen--Cahn energy.

Established a rigidity theorem for critical points similar to Alexandrov's theorem.

Results are uniform across different phase transition length scales.

Abstract

We consider the isoperimetric problem defined on the whole $\mathbb{R}^n$ by the Allen--Cahn energy functional. For non-degenerate 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's theorem for constant mean curvature boundaries.