Convergence rate, location and $\partial_z^2$ 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 $\mathbb C\mathbb P^n$. 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 $\partial_z^2$ condition.