论文标题
$ \ mathbb {h}^2 \ times \ mathbb {r} $和krust属性中的常数平均曲率表面的构造
A construction of constant mean curvature surfaces in $\mathbb{H}^2\times\mathbb{R}$ and the Krust property
论文作者
论文摘要
我们展示了一个$ 2 $ - 参数族的适当的alexandrov包裹的表面，其平均曲率$ 0 \ leq h \ leq h \ leq \ frac {1} {2} {2} $ in $ {\ mathbb {\ mathbb {h}它们相对于水平切片和$ K $垂直飞机对称地处置，并扩展所谓的最小马鞍塔和$ k $ noids。我们表明，当$ h> 0 $分析其偶联的最小表面时，方向起着基本的作用。我们还发现了我们称为$（H，K）$ - Nodoids的新的完整示例，其$ k $端是渐近圆柱体渐近圆柱的曲线，从凸面侧面的曲线$ 2H $，通常会引起非填充的示例，如果$ h> 0 $。 在讨论构造示例的嵌入性时，我们证明了krust属性在任何$ h> 0 $中都不成立，即，在$ \ widetilde {\ mathrm {sl}}} _ 2（\ mathbb {r}）$，$ \ mathers $，$ \ nil}的convex域上的图最少。 $ \ mathbb {h}^2 \ times \ mathbb {r} $ in $ \ mathbb {h}^2^in n n n n n n n n n nograph in $ \ mathbb {h}^2 \ times \ times \ times \ times \ times \ times \ time n n n n n n n n n n n n n nobrage in $ \ mathbb {h}^2 \ times \ times \ times \ times \ times \ times \ time nogrape no graphs in graphs。
We show the existence of a $2$-parameter family of properly Alexandrov-embedded surfaces with constant mean curvature $0\leq H\leq\frac{1}{2}$ in ${\mathbb{H}^2\times\mathbb{R}}$. They are symmetric with respect to a horizontal slice and a $k$ vertical planes disposed symmetrically, and extend the so called minimal saddle towers and $k$-noids. We show that the orientation plays a fundamental role when $H>0$ by analyzing their conjugate minimal surfaces in $\widetilde{\mathrm{SL}}_2(\mathbb{R})$ or $\mathrm{Nil}_3$. We also discover new complete examples that we call $(H,k)$-nodoids, whose $k$ ends are asymptotic to vertical cylinders over curves of geodesic curvature $2H$ from the convex side, often giving rise to non-embedded examples if $H>0$. In the discussion of embeddedness of the constructed examples, we prove that the Krust property does not hold for any $H>0$, i.e., there are minimal graphs over convex domains in $\widetilde{\mathrm{SL}}_2(\mathbb{R})$, $\mathrm{Nil}_3$ or the Berger spheres, whose conjugate surfaces with constant mean curvature $H$ in $\mathbb{H}^2\times\mathbb{R}$ are not graphs.
