Conformal metrics with constant $Q$-curvature for manifolds with boundary

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.

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Published 1 January 2008