On minimal hypersurfaces with finite harmonic indices
On minimal hypersurfaces with finite harmonic indices
复制标题
在具有有限调和指数的最小超曲面上
DOI：
10.1215/s0012-7094-01-11021-1
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发表时间：
2001
影响因子：
2.5
通讯作者：
Senlin Xu
中科院分区：
文献类型：
--
作者：
J. Mei;Senlin Xu
对Rn +1（n3）中的完备极小超曲面引入调和稳定性和调和指数的概念，证明了当调和指数有限时，该超曲面只有n个端点.此外，端部的数量从上到下由1加上谐波指数来限定。每一端都有一个非负调和函数的表示，这些函数构成了单位的一个划分。对于一类特殊的极小超曲面，即具有有限全量曲率的极小超曲面，我们给出了调和指数的一个显式估计。证明了对于这样的子流形，有界调和函数空间是由端点的表示函数生成的。
We introduce the concepts of harmonic stability and harmonic index for a complete minimal hypersurface in R n+1 (n 3) and prove that the hypersurface has only finitely many ends if its harmonic index is finite. Furthermore, the number of ends is bounded from above by 1 plus the harmonic index. Each end has a representation of nonnegative harmonic function, and these functions form a partition of unity. We also give an explicit estimate of the harmonic index for a class of special minimal hypersurfaces, namely, minimal hypersurfaces with finite total scalar curvature. It is shown that for such a submanifold the space of bounded harmonic functions is exactly generated by the representation functions of the ends.