The structure of weakly stable minimal hypersurfaces

Abstract

In this short communication, we announce results from our research on the structure of complete noncompact oriented weakly stable minimal hypersurfaces in a manifold of nonnegative sectional curvature. In particular, a complete oriented weakly stable minimal hypersurface in R m , m ≥ 4, must have only one end; any complete noncompact oriented weakly stable minimal hypersurface has only one end if the complete oriented ambient manifold N m , m ≥ 7, has nonnegative sectional curvature and Ricci curvature bounded below by a positive constant; a complete oriented weakly stable minimal hypersurface in R m , m ≥ 4, with fi nite total scalar curvature is a hyperplane.

Stable complete noncompact hypersurfaces with constant mean curvature

“…Note that Theorems 1 and 2 extend Cheng et al's work ( [7]), since they proved that if N n+1 is a hyperbolic space, n = 3, 4, with H 2 ≥ 10 9 , 7 4 , respectively, M n has only one end. If N n+1 is R n+1 , n = 5, then M 5 has only one end.…”

“…2, the vanishing condition (3) is crucial to study stable hypersurface, i.e., it is necessary to construct a piecewise smooth compactly supported function φ satisfying the vanishing condition (3). The author extends the study of Cheng et al [7] as follows.…”

“…Let M n be a complete, noncompact, stable H -hypersurface in N n+1 . According to [7] Proposition 2.1, it was proved that if E is an end of M n , then…”

Bernstein type theorems for complete submanifolds in space forms

“…In 10, Cheung and Zhou showed that all complete stable hypersurfaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^{n+1}$\end{document} (or \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {H}^{n+1}\big )$\end{document} ( n = 3, 4, 5) with constant mean curvature H > 0(or H > 1, respectively) and ∫ M |ϕ| 2 < +∞ are compact geodesic spheres. Cheng, Cheung and Zhou 8, 9 showed that any complete noncompact weakly stable minimal hypersurfaces must have only one end. 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.…”