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 in Rn. Using an indirect blow-up 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.