Constant T-curvature conformal metrics on 4-manifolds with boundary

Details Constant T-curvature conformal metrics on 4-manifolds with boundary Constant T-curvature conformal metrics on 4-manifolds with boundary

2007-08-06

arxiv.org/abs/0708.0732v1

Mathematics

Analysis of PDEs

Scientific paper

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 one. It has a variational structure, but since the corresponding Euler-Lagrange functional is in general unbounded from below, we look for saddle points. In order to do this, we use topological arguments and min-max methods combined with a compactness result for the corresponding BVP.

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