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In this paper, 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 5 6 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 their quotients. Furthermore, we also classify Einstein manifolds with positive intersection form and an upper bound on the sectional curvature.
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This content will become publicly available on June 12, 2026
Asymptotic behaviour of unstable perturbations of the Fubini–Study metric in Ricci flow
Abstract Kröncke has shown that the Fubini–Study metric is an unstable generalised stationary solution of Ricci flow (Kröncke 2020Commun. Anal. Geom.2835–394). In this paper, we carry out numerical simulations which indicate that Ricci flow solutions originating at unstable perturbations of the Fubini–Study metric develop local singularities modelled by the blowdown soliton discovered in (Feldmanet al2003J. Differ. Geom.65169–209).
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- Award ID(s):
- 2102914
- PAR ID:
- 10639647
- Publisher / Repository:
- IOP Publishing
- Date Published:
- Journal Name:
- Nonlinearity
- Volume:
- 38
- Issue:
- 7
- ISSN:
- 0951-7715
- Page Range / eLocation ID:
- 075015
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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