论文标题
在紧凑的嵌入Weingarten Hypersurfaces上
On compact embbeded Weingarten hypersurfaces in warped products
论文作者
论文摘要
我们表明，紧凑的嵌入式星形$ r $ -R $ -CONVEX超出曲面的某些扭曲产品满足$ H_R = AH+B $，带有$ a \ geqslant 0 $，$ b> 0 $，其中$ h $和$ h $和$ h_r $分别是平均曲率和$ r $ - $ r $ - the Mean-th Mee curvature是一个slice。在空间形式的情况下，我们表明没有星形的假设，这种魏因丁超曲面是地球球体。最后，我们证明，在空间形式的情况下，如果$ hr-ah-b $接近$ 0 $，那么Hypersurface就接近Hausdorff距离的大地测量球。我们还证明了欧几里得空间中这种稳定性结果的各向异性版本。
We show that compact embedded starshaped $r$-convex hypersurfaces of certain warped products satisfying $H_r=aH+b$ with $a\geqslant 0$, $b>0$, where $H$ and $H_r$ are respectively the mean curvature and $r$-th mean curvature is a slice. In the case of space forms, we show that without the assumption of starshapedness, such Weingarten hypersurfaces are geodesic spheres. Finally, we prove that, in the case of space forms, if $Hr-aH-b$ is close to $0$ then the hypersurface is close to geodesic sphere for the Hausdorff distance. We also prove an anisotropic version of this stability result in the Euclidean space.
