\(L^2\) harmonic forms on a complete stable hypersurfaces with constant mean curvature (Q1267466)

The author extends a previous result of \textit{S. Tanno} for minimal hypersurfaces [J. Math. Soc. Japan 48, 761-768 (1996; Zbl 0890.53049)]. He proves the following result. Theorem. Let \(M\) be an n-dimensional (\(2\leqslant n\leqslant 5\)) complete and noncompact (orientable) hypersurface with constant mean curvature \(H\) in a Riemannian manifold of nonnegative bi-Ricci curvature. If \(M\) is strongly stable, then there are no nontrivial \(L^2\) harmonic \(1\)-forms on \(M\). The reviewer remarks that the notion of bi-Ricci curvature in the statement follows \textit{Y. Shen} and \textit{R. Ye} [Duke Math. J. 85, 109-116 (1996; Zbl 0874.53049)]. On the other hand, the concept of strong stability is due to \textit{J. L. Barbosa} and \textit{M. P. do Carmo} [Math. Z. 185, 339-353 (1984; Zbl 0529.53006)].