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 $d\le 7$. 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 $4\le n\le 7$. 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.