On isolated singularities of the conformal Gaussian curvature equation and $Q$-curvature equation

TL;DR

This paper analyzes the behavior of solutions near isolated singularities for the conformal Gaussian curvature and $Q$-curvature equations, extending known results to more general curvature functions and higher dimensions using PDE methods.

Contribution

It provides a PDE-based characterization of isolated singularities for the conformal Gaussian curvature and $Q$-curvature equations, generalizing previous complex analysis approaches.

Findings

Established asymptotic behavior near singularities for nonnegative bounded curvature functions.

Extended analysis to the conformal $Q$-curvature equation in higher dimensions.

Provided a PDE method applicable to general curvature functions, not just constant curvature.

Abstract

In this paper, we study the isolated singularities of the conformal Gaussian curvature equation \[ -\Delta u = K(x) e^{u} \quad ~ in ~ B_{1} \setminus \{ 0 \}, \] where $B_1 \setminus \{ 0 \} \subset \mathbb{R}^2$ is the punctured unit disc. Under the assumption that the Gaussian curvature $K \in L^\infty(B_1)$ is nonnegative, we establish the asymptotic behavior of solutions near the singularity. When $K \equiv 1$, a similar result has been obtained by Chou and Wan (Pacific J. Math. 1994) using the method of complex analysis. Our proof is entirely based on the PDE method and applies to the general Gaussian curvature $K(x)$. Furthermore, our approach is also available for characterizing isolated singularities of the conformal $Q$-curvature equation $(-\Delta)^{\frac{n}{2}} u = K(x) e^{u}$ in any dimension $n\geq 3$. This equation arises from the prescribing $Q$-curvature problem.