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

TL;DR

The paper develops a PDE-centered framework to characterize isolated singularities for the conformal Gaussian curvature equation in the punctured disk and extends the analysis to the conformal $Q$-curvature equation in all dimensions. It introduces a representation formula that separates the singular part from a regular part, enabling precise asymptotics: near the singularity, solutions behave like $u(x) = \alpha \log|x| + \varphi(x)$ with dimension-dependent lower bounds on $\alpha$ and Hölder regular remainder, under a finite-energy condition and nonnegativity of $K$. The results hold for general nonnegative $K \in L^\infty$, including nonlocal odd-dimensional cases, and unify the Gaussian curvature and higher-order $Q$-curvature problems without assuming smoothness or flatness of $K$. This extends known constant-curvature results (e.g., $K\equiv1$) to general $K$ and provides a PDE-based alternative to complex-analytic approaches, with potential impact on prescribed $Q$-curvature problems in geometric analysis.

Abstract

In this paper, we study the isolated singularities of the conformal Gaussian curvature equation \[ -Δu = K(x) e^{u} \quad ~ in ~ B_{1} \setminus \{ 0 \}, \] 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 $(-Δ)^{\frac{n}{2}} u = K(x) e^{u}$ in any dimension $n\geq 3$. This equation arises from the prescribing $Q$-curvature problem.

Theorems & Definitions (14)

• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.2
• Theorem 1.4
• proof : Proof of Theorem \ref{['jdbs']}
• proof : Proof of Theorem \ref{['m=1th']}
• proof : Proof of Theorem \ref{['mth']}
• Lemma 3.1