Harmonic Forms, Price Inequalities, and Benjamini-Schramm Convergence
Publication
, Journal Article
Cerbo, LFD; Stern, M
September 12, 2019
We study Betti numbers of sequences of Riemannian manifolds which Benjamini-Schramm converge to their universal covers. Using the Price inequalities we developed elsewhere, we derive two distinct convergence results. First, under a negative Ricci curvature assumption and no assumption on sign of the sectional curvature, we have a convergence result for weakly uniform discrete sequences of closed Riemannian manifolds. In the negative sectional curvature case, we are able to remove the weakly uniform discreteness assumption. This is achieved by combining a refined Thick-Thin decomposition together with a Moser iteration argument for harmonic forms on manifolds with boundary.
Duke Scholars
Publication Date
September 12, 2019
Citation
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Cerbo, L. F. D., & Stern, M. (2019). Harmonic Forms, Price Inequalities, and Benjamini-Schramm Convergence.
Cerbo, Luca F Di, and Mark Stern. “Harmonic Forms, Price Inequalities, and Benjamini-Schramm Convergence,” September 12, 2019.
Cerbo LFD, Stern M. Harmonic Forms, Price Inequalities, and Benjamini-Schramm Convergence. 2019 Sep 12;
Cerbo, Luca F. Di, and Mark Stern. Harmonic Forms, Price Inequalities, and Benjamini-Schramm Convergence. Sept. 2019.
Cerbo LFD, Stern M. Harmonic Forms, Price Inequalities, and Benjamini-Schramm Convergence. 2019 Sep 12;
Publication Date
September 12, 2019