Twistorial construction of minimal hypersurfaces

Every almost Hermitian structure $(g,J)$ on a four-manifold $M$ determines a hypersurface ${\mathrm{\Sigma }}_J$ in the (positive) twistor space of $(M,g)$ consisting of the complex structures anti-commuting with $J$. In this note we find the conditions under which ${\mathrm{\Sigma }}_J$ is minimal with respect to a natural Riemannian metric on the twistor space in the cases when $J$ is integrable or symplectic. Several examples illustrating the obtained results are also discussed.