Estimates of bubbling solutions of $SU(3)$ Toda systems at critical parameters-Part 1

Abstract: For regular $SU(3)$ Toda systems defined on Riemann surface, we initiate the study of bubbling solutions if parameters $(\rho_1^k,\rho_2^k)$ are both tending to critical positions: $(\rho_1^k,\rho_2^k)\to (4\pi, 4\pi N)$ or $(4\pi N, 4\pi)$ ($N>0$ is an integer). We prove that there are at most three formations of bubbling profiles, and for each formation we identify leading terms of $\rho_1^k-4\pi$ and $\rho_2^k-4\pi N$, locations of blowup points and comparison of bubbling heights with sharp precision. The results of this article will be used as substantial tools for a number of degree counting theorems, critical point at infinity theory in the future.