Monopole Floer homology and the spectral geometry of three-manifolds
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.
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