论文标题
$ m \ times \ mathbb {r} $中的高阶高阶平均曲率的高度曲面
Hypersurfaces of Constant Higher Order Mean Curvature in $M\times\mathbb{R}$
论文作者
论文摘要
我们认为产品$ M \ times \ times \ mathbb r $具有常数$ r $ - th平均曲率$ h_r \ ge 0 $（称为$ h_r $ -hyperSurfaces），其中$ m $是一种任意的riemannian $ n $ n $ manifold。我们开发了一种构造它们的通用方法，并采用它来制作许多示例$ m，包括所有简单连接的空间形式和双曲线空间$ \ mathbb {h} _ {\ mathbb f}^m $（rank $ 1 $ 1 $ symortric nontagpact类型的对称空间）。我们在$ \ mathbb {h} _ {\ Mathbb f}^m \ times \ times \ times \ times \ mathbb r $和$ \ mathbb s^n \ s^n \ times \ mathbb r $中构建和分类完整的旋转$ h_r（\ ge 0）$ - hypersurfaces。它们包括Spheres，Delaunay-type Annuli，对于$ \ Mathbb {h} _ {\ Mathbb f}^M \ times \ times \ Mathbb r，$整体图。我们还构建和分类$ h_r（\ ge 0）$ - $ \ mathbb {h} _ {\ mathbb f}^m \ times \ times \ mathbb r $，这是抛物线异构体或多碳酸酯翻译。我们通过证明$ \ Mathbb h^n \ times \ Mathbb r $或$ \ Mathbb s^n \ s^n \ times \ Mathbb r $ $ $ $ $ $ $（N \ ge 3）$ a旋转Emphere的Spphere，建立了紧凑，连接和严格凸出$ H_R $ -HYPERFACE，建立了一个jellett-liebmann-type定理。获得这些环境空间的完整$ H_R $ -HYPERFACE的其他独特结果。
We consider hypersurfaces of products $M\times\mathbb R$ with constant $r$-th mean curvature $H_r\ge 0$ (to be called $H_r$-hypersurfaces), where $M$ is an arbitrary Riemannian $n$-manifold. We develop a general method for constructing them, and employ it to produce many examples for a variety of manifolds $M,$ including all simply connected space forms and the hyperbolic spaces $\mathbb{H}_{\mathbb F}^m$ (rank $1$ symmetric spaces of noncompact type). We construct and classify complete rotational $H_r(\ge 0)$-hypersurfaces in $\mathbb{H}_{\mathbb F}^m\times\mathbb R$ and in $\mathbb S^n\times\mathbb R$ as well. They include spheres, Delaunay-type annuli and, in the case of $\mathbb{H}_{\mathbb F}^m\times\mathbb R,$ entire graphs. We also construct and classify complete $H_r(\ge 0)$-hypersurfaces of $\mathbb{H}_{\mathbb F}^m\times\mathbb R$ which are invariant by either parabolic isometries or hyperbolic translations. We establish a Jellett-Liebmann-type theorem by showing that a compact, connected and strictly convex $H_r$-hypersurface of $\mathbb H^n\times\mathbb R$ or $\mathbb S^n\times\mathbb R$ $(n\ge 3)$ is a rotational embedded sphere. Other uniqueness results for complete $H_r$-hypersurfaces of these ambient spaces are obtained.
