Stable complete noncompact hypersurfaces with constant mean curvature

Abstract

This article concerns the structure of complete noncompact stable hypersurfaces M n with constant mean curvature H > 0 in a complete noncompact oriented Riemannian manifold N n+1 . In particular, we show that a complete noncompact stable constant mean curvature hypersurface M n , n = 5, 6, in the Euclidean space must have only one end. Any such hypersurface in the hyperbolic space with H 2 > 65 63 , 175 148 , 41 25 , 671 171 for n = 3, 4, 5, 6, respectively, has only one end.

Bernstein type theorems for complete submanifolds in space forms

“…For the case of complete noncompact weakly stable hypersurfaces with constant mean curvature H ≠ 0 in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^{n+1}$\end{document}, they also showed that for n = 5 such hypersurfaces have only one end. Moreover, Fu and Li 14 and Chen 6 proved that the result also holds for n = 6. While for n = 3, 4, Cheng 7 proved that such hypersurfaces do not exist.…”

“…On the other hand, some scholars study the Bernstein problem for constant mean curvature hypersurfaces in the space forms 6, 9, 10, 14, 15, 17, 19. To state some results, we recall some notations and definitions.…”