The Adjunction Inequality for Weyl-Harmonic Maps

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