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Abstract
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In this paper we prove that, given a compact four-dimensional smooth Riemannian
manifold (M,g) with smooth boundary, there exists a metric conformal to g
with constant T-curvature, zero Q-curvature and zero mean curvature under
generic and conformally invariant assumptions. The problem amounts to
solving a fourth-order nonlinear elliptic boundary value problem (BVP) with
boundary conditions given by a third-order pseudodifferential operator and
homogeneous Neumann operator. It has a variational structure, but since the
corresponding Euler–Lagrange functional is in general unbounded from below,
we look for saddle points. We do this by using topological arguments and
min-max methods combined with a compactness result for the corresponding
BVP.
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Keywords
geometric boundary value problems,
blow-up analysis, variational methods, min-max schemes,
Q-curvature, T-curvature, conformal geometry, topological
methods
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Mathematical Subject Classification
Primary: 35B33, 35J35
Secondary: 53A30, 53C21
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Milestones
Received: 1 August 2007
Revised: 1 October 2008
Accepted: 31 October 2008
Published: 2 March 2009
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