Monopole Floer homology and the spectral geometry of three-manifolds
type: Item
instance of: scholarly article
title: Monopole Floer homology and the spectral geometry of three-manifolds
zbMATH Open document ID: 1472.57032
author: Francesco Lin
published in: Communications in Analysis and Geometry
publication date: 2020-11-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/1705.08817
review text: There is a well-developed theory that examines the spectrum of the Laplace operator on manifolds. Less is known about the spectrum of the Laplace operator on forms. This paper uses Seiberg-Witten theory to derive an upper bound on the first eigenvalue of the Hodge Laplacian on co-exact \(1\)-forms on a wide class of \(3\)-manifolds. This result improves prior results that began with the Seiberg-Witten proof of the adjunction inequality. After establishing this upper bound, it is applied to give a new proof of an inequality first established for hyperbolic manifolds by \textit{J. F. Brock} and \textit{N. M. Dunfield} [Invent. Math. 210, No. 2, 531--558 (2017; Zbl 1379.57023)]. This is an example of a result that follows easily once one applies one key idea. In this case after establishing the standard inequality that implies compactness results for the Seiberg-Witten moduli via a Weitzenböck formula, Lin applies the Bochner formula to the form represented by the quadratic term in the first Seiberg-Witten formula. The result is a very clean and clear proof of this interesting result.
reviewed by: David Auckly
Mathematics Subject Classification ID: 57R58
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 53C07
Mathematics Subject Classification ID: 57K41
zbMATH DE Number: 7268920
zbMATH Keywords: Seiberg-Witten theory
zbMATH Keywords: spectral geometry
MaRDI profile type: Publication
arXiv ID: 1705.08817
OpenAlex ID: W3095687393
DOI: 10.4310/CAG.2020.V28.N5.A2
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description: scientific article