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

我们证明了双曲空间中的极小曲面是f-全纯曲线在半黎曼齐次接触空间SL(2，C)/U(1)中的投影。通过使用适当的环群分裂，对于任何给定的势，我们可以构造SL(2，C)/U(1)中的f-全纯曲线。证明了从f-全纯曲线可以得到平均曲率小于1的非最小常平均曲率曲面。利用这种循环群方法(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。对三维Heisenberg群中的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.