Estimates of bubbling solutions of $SU(3)$ Toda systems at critical parameters-Part 2

TL;DR

This paper analyzes bubbling solutions of regular $SU(3)$ Toda systems on Riemann surfaces at critical parameters, demonstrating conditions that prevent bubble accumulation and establishing a spherical Harnack inequality.

Contribution

It proves that under certain conditions, bubbling solutions near a blowup point satisfy a spherical Harnack inequality, ruling out bubble accumulation phenomena.

Findings

Spherical Harnack inequality holds near blowup points.

Bubble accumulation phenomenon is ruled out under certain conditions.

Results are crucial for applications involving $SU(3)$ Toda systems.

Abstract

In this article we study bubbling solutions of regular $SU(3)$ Toda systems defined on a Riemann surface. There are two major difficulties corresponding to the profile of bubbling solutions: partial blowup phenomenon and bubble accumulation. We prove that when both parameters tend to critical positions, if there is one fully bubbling blowup point, then under one curvature assumption, all the blowup solutions near a blowup point satisfy a spherical Harnack inequality, which completely rules out the bubble-accumulation phenomenon. This fact is crucial for a number of applications.