Global construction of constant mean curvature surfaces in terms of contact geometry and loop groups

我们证明了三维双曲空间中的极小曲面可以作为f-全纯曲线在半黎曼齐性切触空间SL（2，C）/U（1）中的投影而得到。通过适当的环群分裂，对于任意给定的势，我们可以构造SL（2，C）/U（1）中的f-全纯曲线.证明了由f-全纯曲线可以得到平均曲率小于1的非极小常平均曲率曲面。利用这种环群方法（新的DPW方法），我们构造了三维双曲空间中的径向对称常平均曲率曲面。我们在三维海森堡群中对极小平移曲面进行了分类。

We showed that minimal surfaces in hyperbolic 3-space are obtained as projections of f-holomorphic curves in the semi-Riemannian homogeneous contact space SL (2,C)/U(1). By using the appropriate loop group splitting, for any prescribed potential, we can construct f-holomorphic curves in SL (2,C)/U(1). It is shown that non-minimal constant mean curvature surfaces with mean curvature less than 1 can be obtained from f-holomorphic curves. By using this loop group method (new DPW-method), we constructed radially symmetric constant mean curvature surfaces in hyperbolic 3-space. We classified minimal translation surfaces in the 3-dimensional Heisenberg group.