Del Pino–Kowalczyk–Wei conjecture on finite-index Allen–Cahn ends

Let $u:\mathbb{R}^3\to[-1,1]$ be a solution to the Allen–Cahn equation with finite Morse index and $\nabla u(x)\neq0$ outside a bounded set. A level-set component is understood through the zero and nonzero level sets of $u$ outside a sufficiently large ball.

Del Pino–Kowalczyk–Wei conjecture. Outside a large ball, each level set of $u$ has finitely many components, each asymptotic either to a plane or to a catenoid; after a rotation of coordinates, all these components are graphs of functions of the same two variables.

The authors proposed this as a parallel to De Giorgi's conjecture. The paper states that the picture is confirmed under a bounded energy-density assumption, while the unrestricted conjecture remains open in the supplied text.