Phase transitions with bounded index: Parallels to De Giorgi's conjecture

A well-known conjecture of De Giorgi -- motivated by analogy with the Bernstein problem for minimal surfaces -- asserts the rigidity of monotone solutions to the Allen--Cahn equation in $\mathbb{R}^{d+1}$, with $d\leq 7$. We establish close parallels to De Giorgi's conjecture for general solutions of bounded Morse index, far stronger than the minimal surface analogy would suggest: Namely, any finite index solution to the Allen--Cahn equation with bounded energy density in $\mathbb{R}^4$ is one-dimensional, and -- conditionally on the classification of stable solutions -- the same holds for all $4\leq n \leq 7$. As a geometric application, phase transitions with bounded energy and index in closed four-manifolds have smooth transition layers which behave like minimal hypersurfaces. Consequently, phase transitions exhibit a remarkably rigid behaviour in higher dimensions. This is in stark contrast with the 3D case, in which a wealth of nontrivial entire solutions with finite index (and energy density) is conversely known to exist, by work pioneered by Del Pino--Kowalczyk--Wei. The authors conjectured that any such solution must have parallel ends which are either planar or catenoidal, suggesting it as a parallel to De Giorgi's conjecture in this framework. We confirm this picture under the bounded energy density assumption.