Vanishing estimates for fully bubbling solutions of $SU(n+1)$ Toda Systems at a singular source

TL;DR

This paper establishes new vanishing estimates for curvature functions in singular Toda systems on Riemann surfaces, revealing a reflection phenomenon that enhances understanding of blowup behavior in these geometric PDEs.

Contribution

It introduces novel vanishing estimates and uncovers a reflection phenomenon for curvature-related functions in Toda systems with singular sources.

Findings

Proves vanishing estimates for curvature functions at blowup points.

Identifies a reflection phenomenon in the behavior of solutions.

Extends understanding of Toda systems with singular sources.

Abstract

For Gauss curvature equation (or more general Toda systems) defined on two dimensional spaces, the vanishing rate of certain curvature functions on blowup points is a key estimate for numerous applications. However, if these equations have singular sources, very few vanishing estimates can be found. In this article we consider a Toda system with singular sources defined on a Riemann surface and we prove a very surprising vanishing estimates and a reflection phenomenon for certain functions involving the Gauss curvature.