$L^2$ harmonic theory, Seiberg-Witten theory and asymptotics of differential forms

Abstract

We present a pair of open smooth $4$-manifolds that are mutually homeomorphic. One of them admits a Riemannian metric that possesses quasi-cylindricity, and positivity of scalar curvature and of dimension of certain $L^2$ harmonic forms. By contrast, for the other manifold, no Riemannian metric can simultaneously satisfy these properties. Our method uses Seiberg-Witten theory on compact $4$-manifolds and applies $L^2$ harmonic theory on non-compact, complete Riemannian $4$-manifolds. We introduce a new argument to apply Gauge theory, which arises from a discovery of an asymptotic property of the range of the differential.

arXiv admin note: text overlap with arXiv:1807.06741

References in corpus (2)

• Lp-cohomology of symmetric spaces
• Covering monopole map and higher degree in non commutative geometry