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The Conformal Laplacian and Positive Scalar Curvature Metrics on Manifolds with Boundary

We give examples of spin $4$-manifolds with boundary $(M,\partial M)$ such that the boundary $\partial M$ has a positive scalar curvature metric which cannot be extended to a positive scalar curvature metric on $M$ with mean convex boundary. These manifolds have the equivalent analytic property that for any metric $g$ on $M$, the conformal Laplacian on $M$ with appropriate boundary conditions and the conformal Laplacian on $\partial M$ cannot both be positive. The obstruction to the positivity of the conformal Laplacians is given by a real-valued $ξ$-invariant associated to the APS theorem for the twisted Dirac operator. We use analytic techniques related to the prescribed scalar curvature problem in conformal geometry to directly treat metrics which are not a product near the boundary.