Four-manifolds with harmonic 2-forms of constant length

  • Inyoung KimEmail author
Original Paper


It was shown by Seaman that if a compact, connected, oriented, riemannian 4-manifold (Mg) 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 \(\mathbb {CP}_{2}\).


4-manifold Harmonic 2-form Biorthogonal curvature Almost-Kähler 

Mathematics Subject Classification

Primary 53C21 53C20 53C24 53D35 



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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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Korea National University of EducationCheongju-siRepublic of Korea

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