Sharp estimates for fully bubbling solutions of a SU(3) Toda system

In this paper, we obtain sharp estimates of fully bubbling solutions of SU(3) Toda system in a compact Riemann surface. In geometry, the SU(n + 1) Toda system is related to holomorphic curves, harmonic maps or harmonic sequences of the Riemann surface to ℂℙⁿ. In order to compute the Leray-Schcuder degree for the Toda system, we have to obtain accurate approximations of the bubbling solutions. Our main goals in this paper are (i) to obtain a sharp convergence rate, (ii) to completely determine the locations, and (iii) to derive the ∂_z² condition, a unexpected and important geometric constraint.