Total scalar curvature and \(L^2\) harmonic 1-forms on a minimal hypersurface in Euclidean space
Let \(M\), \(n=\dim M\geq 3\), be a complete oriented immersed minimal hypersurface in the Euclidean space \(\mathbb R^{n+1}\). If the total scalar curvature of \(M\) is less than \((1/C_s)^{n/2}\), where \(C_s\) is the Sobolev constant for \(M\), then there are no \(L^2\) harmonic 1-forms on \(M\). Consequently, such a hypersurface contains no nontrivial harmonic functions with finite Dirichlet integral and so it has only one end. This finally implies that \(M\) is a hyperplane.
scientific article; zbMATH DE number 1747813