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

TL;DR

This paper analyzes bubbling solutions of $SU(3)$ Toda systems on Riemann surfaces as parameters approach critical values, classifying formations and precise behaviors of blowup solutions.

Contribution

It identifies the possible bubbling formations and determines detailed asymptotic profiles for solutions near critical parameters in $SU(3)$ Toda systems.

Findings

At most three bubbling formations occur.

Precise asymptotic behavior of parameters and blowup points.

Sharp comparison of bubbling heights and profiles.

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.