Some uniformization problems for a fourth order conformal curvature

Abstract

In this paper, we establish the existence of conformal deformations that uniformize fourth order curvature on 4-dimensional Riemannian manifolds with positive conformal invariants. Specifically, we prove that any closed, compact Riemannian manifold with positive Yamabe invariant and total Q-curvature can be conformally deformed into a metric with positive scalar curvature and constant Q-curvature. For a Riemannian manifold with umbilic boundary, positive first Yamabe invariant and total $(Q,T)$-curvature, it is possible to deform it into two types of Riemannian manifolds with totally geodesic boundary and positive scalar curvature. The first type satisfies $Q\equiv \text{constant},T\equiv 0$ while the second type satisfies $Q\equiv 0,T\equiv \text{constant}$.