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.
Categories: math.AP math.DG