\[rhoto1mle4\].

\[rhotome4\] Suppose $\rho^k\to \rho\in \Gamma_1$ and $\rho=Q$. If all $\rho_i^k-Q_i$ have the same sign, then $$\label{12apr23e10}
\Lambda_I(\rho^k)
=-16\pi^2(\sum_{i\in I} b_i e^{c_i}+o(1))\epsilon_k^2\log \epsilon_k^{-1}.$$ where $$b_i=\frac 14\bigg (\Delta \, \log h_i(p)-2K(p)+8\pi+|\nabla \, \log h_i(p)+8\pi \nabla_1\gamma(p,p)|^2\bigg ),$$ $c_i$ is determined in (\[12jun29e1\]).

Important information on bubbling solutions can be observed on the two cases: $\rho\neq Q$ and $\rho=Q$. Theorem \[rhoto1mle4\], which is on $\rho\neq Q$, has its leading term in $\Lambda_I(\rho^k)$ involved with global information of the manifold, while the leading term in Theorem \[rhotome4\], which corresponds to $\rho=Q$, only depends on the geometric information at the blowup point. Moreover, the error terms in Theorem \[rhoto1mle4\] and Theorem \[rhotome4\] respectively also indicate the different asymptotic behaviors of blowup solutions near the singularity. All these differences in the two cases will lead to separate strategies in the construction of bubbling solutions in forthcoming works.

Since Liouville systems and Toda systems share a lot of common features, it is informative to compare our main theorems with the ones for the $SU(3)$ Toda system. First, the location of the blowup point in Theorem \[locations\] is a critical point of a combination of $\log h_i$, $\rho_i$ and $\gamma$ ($\nabla_1 \gamma$ vanishes if the Riemann surface has 