论文标题
与兰瓜最小表面相关的Weingarten表面
Weingarten Surfaces Associated to Laguerre Minimal Surfaces
论文作者
论文摘要
在作品中，作者表明，欧几里得空间中的每个高表面都通过球体的一致性在本地与单位球相关联，该球体的半径函数$ r $是高度的几何不变。在本文中，我们为任何表面$σ$定义其球形平均曲率$ h_s $，取决于$σ$的主曲率和半径函数$ r $。然后，我们考虑两个类别的表面：$ h_s = 0 $的表面，称为$ h_1 $ surfaces，以及带有谐波类型的球形平均曲率的表面，称为$ h_2 $ surfaces。我们为每个类提供了weierstrass-type表示，具体取决于三个全态函数，我们证明$ h_1 $ - 曲面与最小表面相关，而$ h_2 $ -surfaces与laguerre的最小表面有关。作为应用程序，我们为Laguerre最小表面（尤其是最小表面）提供了新的Weierstrass-Type表示形式，以使相同的全体形态数据提供$ H_1 $ -SUR-SURMORTHIC数据以$ H_1 $ -surface/minimalface/minimal表面类别或$ H_2 $ -SUR_2 $ - SURFACE/LAGUERFACE/LAGUERRE Minimal Surface Class类。我们还表征了旋转案例，这使我们能够找到完整的旋转laguerre最小表面。
In the work \cite{Laredo} the author shows that every hypersurface in Euclidean space is locally associated to the unit sphere by a sphere congruence, whose radius function $R$ is a geometric invariant of hypersurface. In this paper we define for any surface $Σ$ its spherical mean curvature $H_S$ which depends on principal curvatures of $Σ$ and the radius function $R$. Then we consider two classes of surfaces: the ones with $H_S = 0$, called $H_1$-surfaces, and the surfaces with spherical mean curvature of harmonic type, named $H_2$-surfaces. We provide for each these classes a Weierstrass-type representation depending on three holomorphic functions and we prove that the $H_1$-surfaces are associated to the minimal surfaces, whereas the $H_2$-surfaces are related to the Laguerre minimal surfaces. As application we provide a new Weierstrass-type representation for the Laguerre minimal surfaces - and in particular for the minimal surfaces - in such a way that the same holomorphic data provide examples in $H_1$-surface/minimal surface classes or in $H_2$-surface/Laguerre minimal surface classes. We also characterize the rotational cases, what allow us finding a complete rotational Laguerre minimal surface.
