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

Abstract

For all rank two Toda systems with an arbitrary singular source, we use a unified approach to prove: (i) The pair of local masses $(σ_1,σ_2)$ at each blowup point has the expression $$σ_i=2(N_{i1}μ_1+N_{i2}μ_2+N_{i3}),$$ where $N_{ij}\in\mathbb{Z},~i=1,2,~j=1,2,3.$ (ii) Suppose at each vortex point $p_t$, $(α_1^t,α_2^t)$ are integers and $ρ_i\notin 4π\mathbb{N}$, then all the solutions of Toda systems are uniformly bounded. (iii) If the blow up point $q$ is not a vortex point, then $$u^k(x)+2\log|x-x^k|\leq C,$$ 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 $α_t^1,α_t^2$ and $1$ are linearly independent over $Q$, then $$u^k(x)+2\log|x-p_t|\leq C.$$ The Harnack type inequalities of (iii) or (iv) is important for studying the bubbling behaves near each blow up point.