Degree counting formula for non-symmetric Toda systems of rank two

Abstract

In this paper, we consider B2 and G2 Toda systems on a compact Riemann surface M. We investigate the relation between the topological property of M and the Leray-Schauder degree of the Toda systems by computing the degree jump caused by multi-bubbling phenomena of the Toda system with non-symmetric Cartan matrix. The bubbling phenomena can reduce the computations of this degree jump to the calculations of the topological degree for some mean field equation together with an additional condition, which is called the shadow system in this paper. The main purpose of this paper is to compute the topological degree of this shadow system via a suitable deformation from a “decoupled” shadow system to our shadow system, and to show the a priori bound along this deformation. The proof of a priori bounds is very subtle, which depends on the type of Lie algebra.