Communications in Analysis and Geometry

Volume 30 (2022)

Number 2

Price inequalities and Betti number growth on manifolds without conjugate points

Pages: 297 – 334

DOI: https://dx.doi.org/10.4310/CAG.2022.v30.n2.a3

Authors

Luca F. Di Cerbo (Department of Mathematics, University of Florida, Gainesville, Fl., U.S.A.)

Mark Stern (Department of Mathematics, Duke University, Durham, North Carolina, U.S.A.)

Abstract

We derive Price inequalities for harmonic forms on manifolds without conjugate points and with a negative Ricci upper bound. The techniques employed in the proof work particularly well for manifolds of non-positive sectional curvature, and in this case we prove a strengthened Price inequality. We employ these inequalities to study the asymptotic behavior of the Betti numbers of coverings of Riemannian manifolds without conjugate points. Finally, we give a vanishing result for $L^2$-Betti numbers of closed manifolds without conjugate points.

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The first-named author was partially supported by a grant associated to the S. S. Chern position at ICTP.

The second-named author was partially supported by Simons Foundation Grant 3553857.

Received 16 May 2018

Accepted 9 August 2019

Published 29 November 2022