四维球面上的稳定超曲面

Stable Hypersurfaces in a 4-Dimensional Sphere

中文关键词:  常纯量曲率  空间形式中的1-极小稳定超曲面

英文关键词:constant scalar curvature  1-minimal stable hypersurfaces in space forms

中文摘要:

      我们研究四维球面中的1-极小稳定完备非紧超曲面.得到在适当限制平均曲率与高斯曲率的情况下,四维球面中不存在具有多项式体积增长的1-极小稳定完备非紧超曲面.这些结果部分证实了外围空间是四维球面时Alencar等人提出的猜测.

英文摘要:

      We study complete noncompact $1$-minimal stable hypersurfaces in a $4$-dimensional sphere $\mathbb{S}^{4}$. We show that there is no complete noncompact $1$-minimal stable hypersurfaces in $\mathbb{S}^{4}$ with polynomial volume growth and the restriction of the mean curvature and Gauss-Kronecker curvature. These results are partial answers to the conjecture of Alencar, do Carmo and Elbert when the ambient space is a $4$-dimensional sphere.