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

我们对自对偶爱因斯坦流形中的零属曲面进行了分类。利用这一结果，我们得到了在特殊情况下用共形变换对所有扭转或全纯曲面空间的商空间的分类。此外，我们还证明了当环境空间为非负标量曲率的自对偶爱因斯坦流形时，扭转或全纯曲面的扭量提升作为调和截面是弱稳定的。反之，如果环境空间为欧几里得空间，则其扭转或升力为弱稳定谐波截面的曲面为扭转或全纯曲面。

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.