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 $({\sigma }_1,{\sigma }_2)$ 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 $$u^k(x)+2\log |x-x^k|\le C,$$ 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 $$u^k(x)+2\log |x-p_t|\le C.$$ The Harnack type inequalities of (iii) or (iv) is important for studying the bubbling behaves near each blow up point.

26 pages

Cited by in corpus (1)

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