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BY 4.0 license Open Access Published by De Gruyter Open Access March 25, 2020

The Adjunction Inequality for Weyl-Harmonic Maps

  • Robert Ream EMAIL logo
From the journal Complex Manifolds


In this paper we study an analog of minimal surfaces called Weyl-minimal surfaces in conformal manifolds with a Weyl connection (M4, c, D). We show that there is an Eells-Salamon type correspondence between nonvertical 𝒥-holomorphic curves in the weightless twistor space and branched Weyl-minimal surfaces. When (M, c, J) is conformally almost-Hermitian, there is a canonical Weyl connection. We show that for the canonical Weyl connection, branched Weyl-minimal surfaces satisfy the adjunction inequality


The ±J-holomorphic curves are automatically Weyl-minimal and satisfy the corresponding equality. These results generalize results of Eells-Salamon and Webster for minimal surfaces in Kähler 4-manifolds as well as their extension to almost-Kähler 4-manifolds by Chen-Tian, Ville, and Ma.

MSC 2010: 32Q60; 53C28; 53C43


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Received: 2019-09-20
Accepted: 2020-03-05
Published Online: 2020-03-25

© 2020 Robert Ream, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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