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

In this paper, we continue to consider the generalized Liouville system: $$\begin{aligned} \Delta _g u_i+\sum _{j=1}^n a_{ij}\rho _j\left( \frac{h_j e^{u_j}}{\int _M h_j e^{u_j}dV_g}- {1} \right) =0\quad \text {in }M,\quad i\in I=\{1,\ldots ,n\}, \end{aligned}$$ where (M, g) is a Riemann surface M with volume 1, $$h_1,\ldots ,h_n$$ are positive $$C^3$$ functions, $$dV_g$$ form, and $$\rho _j\in {\mathbb {R}}^+$$ ( $$j\in I$$ ). previous works Lin-Zhang identified family of hyper-surfaces $$\Gamma _N$$ proved priori estimates for =(\rho _1,\ldots ,\rho _n)$$ in areas separated by . Later also calculated leading term ^k-\rho $$ \in \Gamma _1$$ limit ^k$$ on parameter bubbling sequence. This particularly important applications but it very hard be if tends higher order hypersurface $$N>1$$ Over years numerous attempts have failed article overcome all stumbling blocks completely solve problem under most general context: We not only capture terms , reveal new robust relations coefficient functions at different blowup points.