On Rank Two Toda System with Arbitrary Singularities: Local Mass and New Estimates

For all rank two Toda systems with an arbitrary singular source, we use a unified approach to prove: (i) The pair of local masses $({\sigma }_1,{\sigma }_2)$ at each blowup point has the expression $${\sigma }_i=2(N_{i1}{\mu }_1+N_{i2}{\mu }_2+N_{i3}),$$ where $N_{ij}\in \mathbb{Z},i=1,2,j=1,2,3.$ (ii) Suppose at each vortex point $p_t$, $({\alpha }_1^t,{\alpha }_2^t)$ are integers and ${\rho }_i\mathrm{\notin }4\pi \mathbb{N}$, then all the solutions of Toda systems are uniformly bounded. (iii) If the blow up point $q$ is not a vortex point, then $$u^k(x)+2\log \mathrm{\mid }x-x^k\mathrm{\mid }\le C,$$ where $x^k$ is the local maximum point of $u^k$ near $q$. (iv) If the blow up point $q$ is a vortex point $p_t$ and ${\alpha }_t^1,{\alpha }_t^2$ and $1$ are linearly independent over $Q$, then $$u^k(x)+2\log \mathrm{\mid }x-p_t\mathrm{\mid }\le C\mathrm{.}$$ The Harnack type inequalities of (iii) or (iv) is important for studying the bubbling behaves near each blow up point.