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

Abstract

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.