Abstract
It was shown by Seaman that if a compact, connected, oriented, riemannian 4-manifold (M, g) of positive sectional curvature admits a harmonic 2-form of constant length, then M has definite intersection form and such a harmonic form is unique up to constant multiples. In this paper, we show that such a manifold is diffeomorphic to \(\mathbb {CP}_{2}\) with a slightly weaker curvature hypothesis and there is an infinite dimensional moduli space of such metrics near the Fubini-Study metric on \(\mathbb {CP}_{2}\).
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Acknowledgements
The author is very thankful to Prof. Claude LeBrun for suggesting the problem and helpful discussions. The author is grateful to the referee who suggested a better proof of Lemma 3, another proof of Theorem 4 in case of positive sectional curvature and introduced the use of a function \(\kappa \) with other careful suggestions. This article was supported by NRF-2018R1D1A3B07043346. The author would like to thank Chanyoung Sung and Korea National University of Education.
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Kim, I. Four-manifolds with harmonic 2-forms of constant length. Geom Dedicata 207, 209–218 (2020). https://doi.org/10.1007/s10711-019-00494-6
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DOI: https://doi.org/10.1007/s10711-019-00494-6