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On Liouville systems at critical parameters, Part 2: Multiple bubbles

In this paper, we continue to consider the generalized Liouville system: $$ Δ_g u_i+\sum_{j=1}^n a_{ij}ρ_j\left(\frac{h_j e^{u_j}}{\int h_j e^{u_j}}- {1} \right)=0\quad\text{in \,}M,\quad i\in I=\{1,\cdots,n\}, $$ where $(M,g)$ is a Riemann surface $M$ with volume $1$, $h_1,..,h_n$ are positive smooth functions and $ρ_j\in \mathbb R^+$($j\in I$). In previous works Lin-Zhang identified a family of hyper-surfaces $Γ_N$ and proved a priori estimates for $ρ=(ρ_1,..,ρ_n)$ in areas separated by $Γ_N$. Later Lin-Zhang also calculated the leading term of $ρ^k-ρ$ where $ρ\in Γ_1$ is the limit of $ρ^k$ on $Γ_1$ and $ρ^k$ is the parameter of a bubbling sequence. This leading term is particularly important for applications but it is very hard to be identified if $ρ^k$ tends to a higher order hypersurface $Γ_N$ ($N>1$). Over the years numerous attempts have failed but in this article we overcome all the stumbling blocks and completely solve the problem under the most general context: We not only capture the leading terms of $ρ^k-ρ\in Γ_N$, but also reveal new robustness relations of coefficient functions at different blowup points.