Complete Hypersurfaces with Bounded Mean Curvature in ${\mathbb R}^{n+1}$

Wang, Qiaoling; Xia, Changyu

Bull. Belg. Math. Soc. Simon Stevin, Tome 13 (2007) no. 5, p. 607-619 / Harvested from Project Euclid

Résumé

Let $M$ be an $n$-dimensional complete non-compact hypersurface in ${\mathbb R}^{n+1}$ and assume that its mean curvature lies between two positive numbers. Denote by $\Delta$ and $A$ the Laplacian operator and the second fundamental form of $M$, respectively. In this paper, we show that if $3\leq n\leq 5$ and if ${\rm Ind}(\Delta +|A|^2)$ is finite, then $M$ has finitely many ends. We also show that if $2\leq n\leq 5$ and if ${\rm Ind }(\Delta +|A|^2)=0$, then $M$ has only one end.

Publié le : 2007-11-14
Classification: Complete hypersurfaces, mean curvature, finite index, ends, 53C20, 53C42

@article{1195157130,
     author = {Wang, Qiaoling and Xia, Changyu},
     title = {Complete Hypersurfaces with Bounded Mean Curvature in ${\mathbb R}^{n+1}$},
     journal = {Bull. Belg. Math. Soc. Simon Stevin},
     volume = {13},
     number = {5},
     year = {2007},
     pages = { 607-619},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1195157130}
}

Wang, Qiaoling; Xia, Changyu. Complete Hypersurfaces with Bounded Mean Curvature in ${\mathbb R}^{n+1}$. Bull. Belg. Math. Soc. Simon Stevin, Tome 13 (2007) no. 5, pp.  607-619. http://gdmltest.u-ga.fr/item/1195157130/