Finite Morse index implies finite ends

Abstract: We prove that finite Morse index solutions to the Allen-Cahn equation in $\R^2$ have {\bf finitely many ends} and {\bf linear energy growth}. The main tool is a {\bf 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 $\R^n$. Using an indirect blow-up technique, in the spirit of the classical Colding-Minicozzi theory in minimal surfaces, we show that the {\bf obstruction} to the uniform second order regularity of clustering interfaces in $\R^n$ is associated to the existence of nontrivial entire solutions to a (finite or infinite) {\bf Toda system} in $\R^{n-1}$. For finite Morse index solutions in $\R^2$, we show that this obstruction does not exist by using information on stable solutions of the Toda system.