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 $Q\equiv 0, T \equiv \text{constant}$.