On Rank-2 Toda Systems with Arbitrary Singularities: Local Mass and New Estimates
Journal
Analysis & Pde
Journal Volume
11
Journal Issue
4
Pages
873-898
Date Issued
2018
Author(s)
Abstract
For all rank-2 Toda systems with an arbitrary singular source, we use a unified approach to prove:(1) The pair of local masses . 1 ; 2 / at each blowup point has the expressionwhere N ij 2 , i D 1; 2, j D 1; 2; 3.(2) At each vortex point p t if .1t ; 2 t / are integers and i 4 , then all the solutions of Toda systems are uniformly bounded.(3) If the blowup point q is a vortex point p t and 1 t ; 2 t and 1 are linearly independent over Q, thenThe Harnack-type inequalities of 3 are important for studying the bubbling behavior near each blowup point. where g is the Laplace-Beltrami operator ( g 0), S is a finite set on M, h 1 ; : : : ; h n are positive and smooth functions on M, i t > 1 is the strength of the Dirac mass p t and D . 1 ; : : : ; n / is a constant vector with nonnegative components.Here for simplicity we just assume that the total area of M is 1.