Kazdan-Warner obstructions for a 4$th-$order boundary problem

TL;DR

This paper establishes Kazdan-Warner type identities for a boundary value problem involving prescribing interior $Q$-curvature and boundary $T$-curvature on the 4-dimensional upper hemisphere, revealing integral obstructions to solutions.

Contribution

It introduces new Kazdan-Warner type identities specific to a 4th-order boundary problem, providing a novel approach to understanding solvability conditions.

Findings

Derived integral obstructions to the boundary problem's solvability.

Identified conformal invariance properties related to the problem.

Established variational formulations using boundary-preserving conformal vector fields.

Abstract

We derive Kazdan-Warner type identities for the boundary problem of prescribing nonconstant interior $Q$ curvature and boundary $T$ curvature on the upper hemisphere $\mathbb{S}^4_+$ by a conformal change of the standard metric. Using the natural variational formulation and conformal variations generated by boundary-preserving conformal vector fields, we obtain nontrivial integral obstructions to solvability.