Classification theorems for solutions of higher order boundary conformally invariant problems, I

TL;DR

This paper classifies solutions to higher order boundary conformally invariant problems, showing they are composed of polynomials and bubbles, with implications for metrics with singular boundary points in conformal geometry.

Contribution

It introduces a novel classification theorem for solutions of higher order boundary problems, revealing their structure as polynomials plus bubbles and analyzing their geometric implications.

Findings

Solutions are polynomials plus bubbles.

Existence of metrics with a single singular boundary point and flat Q-curvature.

Classification of all such metrics.

Abstract

In this paper, we prove that nonnegative polyharmonic functions on the upper half space satisfying a conformally invariant nonlinear boundary condition have to be the "\emph{polynomials} plus \emph{bubbles}" form. The nonlinear problem is motivated by the recent studies of boundary GJMS operators and the $Q$-curvature in conformal geometry. The result implies that in the conformal class of the unit Euclidean ball there exist metrics with a single singular boundary point which have flat $Q$-curvature and constant boundary $Q$-curvature. Moreover, all of such metrics are classified. This phenomenon differs from that of boundary singular metrics which have flat scalar curvature and constant mean curvature, where the singular set contains at least two points. A crucial ingredient of the proof is developing an approach to separate the higher order linear effect and the boundary nonlinear effect so that the kernels of the nonlinear problem are captured.