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

TL;DR

This paper constructs examples of 4-manifolds with boundary where positive scalar curvature metrics on the boundary cannot extend inward, linking geometric obstructions to spectral invariants of the Dirac operator.

Contribution

It provides explicit examples of manifolds with boundary exhibiting obstructions to extending positive scalar curvature metrics, using analytic and conformal geometry techniques.

Findings

Existence of 4-manifolds with boundary with non-extendable positive scalar curvature boundary metrics

Obstruction characterized by a real-valued i-invariant from APS index theory

Analytic approach to scalar curvature problems without product boundary assumptions

Abstract

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 $\xi$-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.