Ribaucour transformations for constant mean curvature and linear Weingarten surfaces
Ribaucour transformations for constant mean curvature and linear Weingarten surfaces
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恒定平均曲率和线性 Weingarten 曲面的 Ribaucour 变换
DOI：
10.2140/pjm.2003.212.265
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发表时间：
2003
影响因子：
0.6
通讯作者：
K. Tenenblat
中科院分区：
文献类型：
--
作者：
A. Corro;W. Ferreira;K. Tenenblat
我们提供了一种方法，通过在 Ribaucour 变换上施加单参数代数条件，从给定的此类曲面获得线性 Weingarten 曲面。我们的主要结果扩展了恒定高斯或平均曲率表面的经典结果。通过将该理论应用于圆柱体，我们获得了一个二参数族的完整线性温加滕曲面（双曲、椭圆和管状），渐近地接近圆柱体，当其中一个参数消失时，它们具有恒定的平均曲率。该族包含 n 泡 Weingarten 曲面，其为 1 周期，具有亏格零和几何索引 m 的两端，其中 n/m 是不可约有理数。它们的总曲率消失，而总绝对曲率是 8πn。我们还应用该方法来获得与 Delaunay 曲面相关的完整常平均曲率曲面族，这些曲面对于参数的特殊值是 1-周期的。
We provide a method to obtain linear Weingarten surfaces from a given such surface, by imposing a one parameter algebraic condition on a Ribaucour transformation. Our main result extends classical results for surfaces of constant Gaussian or mean curvature. By applying the theory to the cylinder, we obtain a two-parameter family of complete linear Weingarten surfaces (hyperbolic, elliptic and tubular), asymptotically close to the cylinder, which have constant mean curvature when one of the parameters vanishes. The family contains n-bubble Weingarten surfaces which are 1-periodic, have genus zero and two ends of geometric index m, where n/m is an irreducible rational number. Their total curvature vanishes, while the total absolute curvature is 8πn. We also apply the method to obtain families of complete constant mean curvature surfaces, associated to the Delaunay surfaces, which are 1-periodic for special values of the parameter.