Hokkaido Mathematical Journal

Hokkaido Mathematical Journal, 40 (2011) pp.103-110

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Abstract

Let Mn (n ≥ 3) be a complete δ (> $\frac{(n-1)^2}{n^2}$)-stable minimal hypersurface in an (n + 1)-dimensional Euclidean space $\mathbb{R}^{n+1}$. We prove that there are no nontrivial $L^2$ harmonic 1-forms on M and the first de Rham's cohomology group with compact support of M is trivial. As corollaries, M has only one end. This implies that if M has finite total curvature, then M is a hyperplane.

MSC(Primary) 53C42 end, $L^2$ harmonic forms, minimal hypersurface, stability