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

TL;DR

This paper investigates the behavior of solutions to rank two Toda systems with arbitrary singularities, deriving formulas for local masses, establishing boundedness under certain conditions, and providing inequalities crucial for analyzing bubbling phenomena.

Contribution

The paper introduces a unified approach to characterize local masses, proves boundedness of solutions under specific conditions, and establishes new inequalities for analyzing blowup behavior.

Findings

Local masses at blowup points are expressed via integer combinations of parameters.

Solutions are uniformly bounded when vortex point parameters are integers and certain conditions are met.

Harnack-type inequalities are established for solutions near blowup points, aiding in bubbling analysis.

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 $$u^k(x)+2\log|x-x^k|\leq C,$$ 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.