Research of surfaces in four-dimensional Riemannian manifolds using their twistor lifts

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We classify surfaces of genus zero in self-dual Einstein manifolds whose twistor lifts are harmonic sections. Using this result, we obtain a classification for the quotient space of the space of all twistor holomorphis surfaces by conformal transformations in the special case. Moreover we show that twsitor lifts of twistor holomorphic surfaces are weakly stable as harmonic sections if ambient spaces are self-dual Einstein manifolds of non-negative scalar curvature. Conversely, if the ambient space is Euclidean space, a surface whose twistor lift is a weakly stable harmonic section is twistor holomorphic.