论文标题
$ \ mathbb {s}^2 \ times \ mathbb {r} $和$ \ mathbb {h}^2 \ times \ times \ times \ mathbb {r} $ in $ \ mathbb {s}^2 \ mathbb {r} $ \ mathbb {r} $
Horizontal Delaunay surfaces with constant mean curvature in $\mathbb{S}^2\times\mathbb{R}$ and $\mathbb{H}^2\times\mathbb{R}$
论文作者
论文摘要
我们在$ \ Mathbb {s}^2 \ times \ Mathbb {r} $和$ \ mathbb {h}^2 \ times \ times \ times \ times \ mathbb {rsbb {r} $中，在$ \ mathbb {s}^2 \ times \ timebb {r} $中，我们获得了$ 1 $ - 参数的水平delaunay表面家族，其平均曲率为正恒定平均值。这些表面不是均等的，而是单一的周期性，位于与水平测量学的有界距离处，并完成了作者先前给出的水平流胶家族。我们详细研究了整个家庭的几何形状，并表明将水平流驱动器正确嵌入$ \ Mathbb H^2 \ Times \ Mathbb {R} $中。我们还发现（在起伏不泄露中）嵌入的常数平均曲率属于$ \ mathbb s^2 \ times \ mathbb {r} $，它们是从一堆切线球到水平不变圆柱体的连续变形。特别是，我们找到了$ \ mathbb {s}^2 \ times \ times \ mathbb {r} $中的第一个非等级示例，它们具有常数的平均曲率$ h> \ frac12 $。最后，我们证明没有适当沉浸的表面，具有恒定的平均曲率$ h \ leq \ frac {1} {2} {2} $在$ \ mathbb {h}^2 \ times \ times \ times \ mathbb {r} $中的水平测量距离的边界距离处。
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 $\mathbb{H}^2\times\mathbb{R}$.
