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 \(\rho_i\notin 4\pi\mathbb{N}\), then all the solutions of Toda systems are uniformly bounded. (iii) If the blow up point \(q\) is not a vortex point, then $$u^k(x)+2\log|x-x^k|\leq C,$$ 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 \(\alpha_t^1,\alpha_t^2\) 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.