Monopole Floer homology and the spectral geometry of three-manifolds

We refine some classical estimates in Seiberg–Witten theory, and discuss an application to the spectral geometry of three-manifolds. We show that for any Riemannian metric on a rational homology three-sphere $Y$, the first eigenvalue of the Hodge Laplacian on coexact one-forms is bounded above explicitly in terms of the Ricci curvature, provided that $Y$ is not an $L$-space (in the sense of Floer homology). The latter is a computable purely topological condition, and holds in a variety of examples. Performing the analogous refinement in the case of manifolds with $b_1 \gt 0$, we provide a gauge-theoretic proof of an inequality of Brock and Dunfield relating the Thurston and $L^2$ norms of hyperbolic three-manifolds, first proved using minimal surfaces.

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This work was supported by the the Shing-Shen Chern Membership Fund, and by the IAS Fund for Mathematics.

Received 16 October 2017

Accepted 30 April 2018

Published 14 October 2020