Communications in Analysis and Geometry
Volume 16 (2008)
Conformal metrics with constant $Q$-curvature for manifolds with boundary
Pages: 1049 – 1124
In this paper we prove that, given a compact four-dimensional smoothRiemannian manifold $(M,g)$ with smooth boundary, there exists ametric in the conformal class [$g$] of the background metric $g$ withconstant $Q$-curvature, zero $T$-curvature and zero mean curvatureunder generic conformally invariant assumptions. The problem isequivalent to solving a fourth-order non-linear elliptic boundaryvalue problem (BVP) with boundary condition given by a third-orderpseudodifferential operator and homogeneous\break Neumann condition. It hasa variational structure, but since the corresponding Euler--Lagrangefunctional is in general unbounded from above and below, we need touse min--max methods combined with a new topological argument and acompactness result for the above~BVP.