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It is well known that the study of $SU(n+1)$ Toda systems is important notonly to Chern-Simons models in Physics, but also to the understanding ofholomorphic curves, harmonic sequences or harmonic maps from Riemann surfacesto $\mathbb C\mathbb P^n$. One major goal in the study of $SU(n+1)$ Toda systemon Riemann surfaces is to completely understand the asymptotic behavior offully bubbling solutions. In this article we use a unified approach to studyfully bubbling solutions to general $SU(n+1)$ Toda systems and we prove threemajor sharp estimates important for constructing bubbling solutions: thecloseness of blowup solutions to entire solutions, the location of blowuppoints and a $\partial_z^2$ condition.

Convergence rate, location and $\partial_z^2$ condition for fully bubbling solutions to SU(n+1) Toda Systems