Classification of solutions to the $Q$-flat and constant $T$-curvature equation on the half-space and ball

TL;DR

This paper introduces a biharmonic Poisson kernel for conformal boundary operators and uses it to classify nonnegative solutions to specific curvature equations on half-space and ball domains.

Contribution

It provides a rigorous definition of the biharmonic Poisson kernel and applies it to classify solutions of curvature equations in geometric analysis.

Findings

Explicit representation formula for the biharmonic Poisson kernel.

Classification theorems for solutions on half-space and ball.

New tools for analyzing conformal curvature equations.

Abstract

For conformal boundary operators associated with the Paneitz operator, we introduce a rigorous definition of the biharmonic Poisson kernel consisting of a pair of kernel functions and derive its explicit representation formula. With this powerful tool, we establish classification theorems of nonnegative solutions to the $Q$-flat and constant $T$-curvature equations on $\mathbb{R}_+^{n+1}$ and $\mathbb{B}^{n+1}$.