Horizontal Delaunay surfaces with constant mean curvature in $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$

IF 1.8 2区 数学 Q1 MATHEMATICS Cambridge Journal of Mathematics Pub Date : 2020-07-14 DOI:10.4310/cjm.2022.v10.n3.a2
J. M. Manzano, Francisco Torralbo
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Abstract

We obtain a $1$-parameter family of horizontal Delaunay surfaces with positive constant mean curvature in $\mathbb{S}^2\times\mathbb{R}$ and $\mathbb{H}^2\times\mathbb{R}$, being the mean curvature larger than $\frac{1}{2}$ in the latter case. These surfaces are not equivariant but singly periodic, lie at bounded distance from a horizontal geodesic, and complete the family of horizontal unduloids previously given by the authors. We study in detail the geometry of the whole family and show that horizontal unduloids are properly embedded in $\mathbb H^2\times\mathbb{R}$. We also find (among unduloids) families of embedded constant mean curvature tori in $\mathbb S^2\times\mathbb{R}$ which are continuous deformations from a stack of tangent spheres to a horizontal invariant cylinder. In particular, we find the first non-equivariant examples of embedded tori in $\mathbb{S}^2\times\mathbb{R}$, which have constant mean curvature $H>\frac12$. Finally, we prove that there are no properly immersed surface with constant mean curvature $H\leq\frac{1}{2}$ at bounded distance from a horizontal geodesic in $\mathbb{H}^2\times\mathbb{R}$.
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具有常数平均曲率的水平Delaunay曲面,单位为$\mathbb{S}^2 \times\mathb{R}$和$\mathbb{H}^2 \times\mathbb{R}$
我们得到了一个水平Delaunay曲面的$1$参数族,其正常平均曲率为$\mathbb{S}^2 \times\mathb{R}$和$\mathbb{H}^2 \times\mathbb{R}$,在后一种情况下,平均曲率大于$\frac{1}{2}$。这些曲面不是等变的,而是单周期的,位于距离水平测地线有界的距离处,并完成了作者先前给出的水平unduloid族。我们详细研究了整个家族的几何结构,并证明了水平unduloid正确地嵌入在$\mathbb H^2 \times\mathbb{R}$中。我们还发现(在unduloid中)$\mathbb S^2 \times\mathbb{R}$中嵌入常平均曲率tori的族,它们是从一堆相切球体到水平不变圆柱体的连续变形。特别地,我们在$\mathbb{S}^2 \times\mathbb{R}$中发现了嵌入环面的第一个非等变例子,它们具有恒定的平均曲率$H>\frac12$。最后,我们证明了在距离$\mathbb{H}^2 \times\mathbb{R}$中的水平测地线有界距离处,不存在具有常平均曲率$H\leq\frac{1}{2}$的适当浸入曲面。
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CiteScore
3.10
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0.00%
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