Mathematical Research Letters

Volume 4 (1997)

Number 5

The structure of stable minimal hypersurfaces in $I\!\!R^{n+1}$

Pages: 637 – 644



Huai-Dong Cao

Ying Shen

Shunhui Zhu


We provide a new topological obstruction for complete stable minimal hypersurfaces in $I\!\!R^{n+1}$. For $n\geq 3$, we prove that a complete orientable stable minimal hypersurface in $I\!\!R^{n+1}$ cannot have more than one end by showing the existence of a bounded harmonic function based on the Sobolev inequality for minimal submanifolds \cite{MS} and by applying the Liouville theorem for harmonic functions due to Schoen-Yau \cite{SY}.

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