L 2-harmonic forms and stable hypersurfaces in space forms

Abstract

We prove that a complete noncompact orientable stable minimal hypersurface in $${\mathbb{S}^{n+1}}$$ (n ≤ 4) admits no nontrivial L 2-harmonic forms. We also obtain that a complete noncompact strongly stable hypersurface with constant mean curvature in $${\mathbb{R}^{n+1}}$$ or $${\mathbb{S}^{n+1}}$$ (n ≤ 4) admits no nontrivial L 2-harmonic forms. These results are generalized versions of Tanno’s result on stable minimal hypersurfaces in $${\mathbb{R}^{n+1}}$$ .