Finite Morse Index Implies Finite Ends

We prove that finite Morse index solutions to the Allen‐Cahn equation in ℝ2 have finitely many ends and linear energy growth. The main tool is a curvature decay estimate on level sets of these finite Morse index solutions, which in turn is reduced to a problem on the uniform second‐order regularity of clustering interfaces for the singularly perturbed Allen‐Cahn equation. Using an indirect blowup technique, in the spirit of the classical Colding‐Minicozzi theory in minimal surfaces, we show that the obstruction to the uniform second‐order regularity of clustering interfaces in ℝn is associated to the existence of nontrivial entire solutions to a (finite or infinite) Toda system in ℝn–1. For finite Morse index solutions in ℝ2, we show that this obstruction does not exist by using information on stable solutions of the Toda system. © 2019 Wiley Periodicals, Inc.