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

Tsuyoshi Kato

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arXiv · 2001.02128

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

Tsuyoshi Kato

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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.

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Tsuyoshi Kato. 2020-01-06. $L^2$ harmonic theory, Seiberg-Witten theory and asymptotics of differential forms. https://arxiv.org/abs/2001.02128

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