Classification and a priori estimates for the singular prescribing $Q$-curvature equation on 4-manifold

TL;DR

This paper studies the prescribed Q-curvature equation on 4-manifolds with singular sources, providing classification, quantization, and a priori estimates, advancing understanding of singular solutions and their blow-up behavior.

Contribution

It offers a classification theorem for singular Liouville equations on 4, analyzes concentration phenomena, and establishes a priori estimates and inequalities for singular solutions.

Findings

Classification of singular Liouville equations on 4

Quantization of bubbling solutions

Spherical Harnack inequality for non-integer singular sources

Abstract

On $(M,g)$ a compact riemannian $4-$manifold we consider the prescribed $Q-$curvature equation defined on $M$ with finite singular sources. We first prove a classification theorem for singular Liouville equations defined on $\mathbb R^4$ and perform a concentration compactness analysis. Then we derive a quantization result for bubbling solutions and establish a priori estimate under the assumption that certain conformal invariant does not take some quantized values. Furthermore we prove a spherical Harnack inequality around singular sources provided their strength is not an integer. Such an inequality implies that in this case singular sources are \emph{isolated simple blow up points}.