The sharp type Chern-Gauss-Bonnet integral and asymptotic behavior

TL;DR

This paper introduces a precise criterion based on $Q$ curvature to evaluate the Chern-Gauss-Bonnet integral and derives asymptotic formulas for solutions to the $Q$ curvature equation, using a novel estimation approach.

Contribution

It presents a new sharp criterion for the Chern-Gauss-Bonnet integral focusing on $Q$ curvature and develops asymptotic formulas using innovative singular integral estimates.

Findings

Established a quantitative criterion based on $Q$ curvature

Derived asymptotic formulas for $Q$ curvature equation solutions

Introduced a new approach involving singular integral estimation

Abstract

In this paper, we propose a sharp and quantitative criterion, which focuses solely on $Q$ curvature, to demonstrate the Chern-Gauss-Bonnet integral. In contrast to the previous results [4,5,10], we use a new approach that involves estimating the singular integral. Furthermore, we derive the asymptotic formula for the solution to the general $Q$ curvature equation.