A priori estimates of Toda systems, I: the Lie algebras of $\mathbf{A}_n$, $\mathbf{B}_n$, $\mathbf{C}_n$ and $\mathbf{G}_2$

Abstract

It is well known that the PDE (Partial Differential Equation) system arising from the infinitesimal Plucker formulas is a particular case of the Toda system of $\mathbf{A}_n$ type. In this paper, we prove an a priori estimate of solutions of the Toda systems associated with the simple Lie algebras $\mathbf{A}_n$, $\mathbf{B}_n$, $\mathbf{C}_n$ and $\mathbf{G}_2$. Previous results in this direction have been done only for the case of Lie algebras of rank two. Our result for $n \geq 3$ is new. The proof of this fundamental theorem is to combine techniques from PDE and the monodromy theory. One of the key steps is to calculate the local mass of a sequence of blowup solutions near each blowup point. At each blowup point, a sequence of bubbling steps (via scaling) are performed, and the local mass of the present step could be computed from the previous step. We find out that this transformation of the local mass of each step is related to the action of an element in the Weyl group of the Lie algebra. The correspondence of the Pohozaev identity and the Weyl group could reduce the complicated calculation of the local mass into a simpler one.