Convergence rate, location and ∂z2 condition for fully bubbling solutions to SU(n+1) Toda systems

It is well known that the study of SU(n+1) Toda systems is important not only to Chern-Simons models in Physics, but also to the understanding of holomorphic curves, harmonic sequences or harmonic maps from Riemann surfaces to CPn. One major goal in the study of SU(n+1) Toda system on Riemann surfaces is to completely understand the asymptotic behavior of fully bubbling solutions. In this article we use a unified approach to study fully bubbling solutions to general SU(n+1) Toda systems and we prove three major sharp estimates important for constructing bubbling solutions: the closeness of blowup solutions to entire solutions, the location of blowup points and a ∂z2 condition.