On rank-2 Toda systems with arbitrary singularities: local mass and new estimates

Abstract

For all rank-2 Toda systems with an arbitrary singular source, we use a unified approach to prove:

1. The pair of local masses $\left({\sigma }_1,{\sigma }_2\right)$ at each blowup point has the expression $${\sigma }_i=2\left(N_{i1}{\mu }_1+N_{i2}{\mu }_2+N_{i3}\right),$$ where $N_{ij}\in \mathbb{Z}$, $i=1,2$, $j=1,2,3$.

2. At each vortex point $p_t$ if $\left({\alpha }_t^1,{\alpha }_t^2\right)$ are integers and ${\rho }_i\notin 4\pi \mathbb{N}$, then all the solutions of Toda systems are uniformly bounded.

3. If the blowup point $q$ is a vortex point $p_t$ and ${\alpha }_t^1,{\alpha }_t^2$ and $1$ are linearly independent over $Q$, then $$u^k\left(x\right)+2log|x-p_t|\le C.$$

The Harnack-type inequalities of 3 are important for studying the bubbling behavior near each blowup point.