Harmonic 1-Forms on Minimal Hypersurfaces in $${{\mathbb {S}}}^{n}\times {{\mathbb {R}}}$$

Abstract

We consider a complete noncompact minimal hypersurface \(\Sigma ^n\) in a product manifold \({{\mathbb {S}}}^{n}(\sqrt{2(n-1)})\times {{\mathbb {R}}}\) \((n\ge 3)\). We get that there admits no nontrivial \(L^2\) harmonic 1-forms on \(\Sigma \) if the square of \(L^n\)-norm of the second fundamental form is less than \(\frac{\alpha ^2n}{2C_0(n-1)}\) or the square of the length of the second fundamental form is less than \(\frac{n\alpha ^2}{2(n-1)}\). Here \(\alpha \) is an angle function and \(C_0\) is the Sobolev constant depending only on n.