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arXiv · 1908.09958

New Curvature Conditions for the Bochner Technique

Peter Petersen·Matthias Wink
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Abstract

We prove a vanishing and estimation theorem for the $p^{\text{th}}$-Betti number of closed $n$-dimensional Riemannian manifolds with a lower bound on the average of the lowest $n-p$ eigenvalues of the curvature operator. This generalizes results due to D. Meyer, Gallot-Meyer, and Gallot. For example, in dimensions $n=5,6$ we obtain vanishing of the Betti numbers provided that the curvature operator is $3$-positive. As B\"ohm-Wilking observed, $3$-positivity of the curvature operator is not preserved by the Ricci flow.

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Peter Petersen, Matthias Wink. 2019-08-26. New Curvature Conditions for the Bochner Technique. https://doi.org/10.1007/s00222-020-01003-3

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