Vol. 319, No. 1, 2022

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Four-manifolds of pinched sectional curvature

Xiaodong Cao and Hung Tran

Vol. 319 (2022), No. 1, 17–38
Abstract

We study closed four-dimensional manifolds. In particular, we show that under various pinching curvature conditions (for example, the sectional curvature is no more than 56 of the smallest Ricci eigenvalue), the manifold is definite. If restricting to a metric with harmonic Weyl tensor, then it must be self-dual or anti-self-dual under the same conditions. Similarly, if restricting to an Einstein metric, then it must be either the complex projective space with its Fubini–Study metric, the round sphere, or the quotient of one of these. Furthermore, we also classify Einstein manifolds with positive intersection form and an upper bound on the sectional curvature.

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Keywords
rigidity, Hopf conjecture, definite, Bochner technique, harmonic Weyl, Einstein
Mathematical Subject Classification
Primary: 53C25
Milestones
Received: 25 July 2020
Revised: 19 November 2020
Accepted: 22 January 2022
Published: 28 August 2022
Authors
Xiaodong Cao
Department of Mathematics
Cornell University
Ithaca, NY
United States
Hung Tran
Department of Mathematics and Statistics
Texas Tech University
Lubbock, TX
United States