Minimal hypersurfaces of a positive scalar curvature manifold
This is a systematic study of the placement question for minimal hypersurfaces in a manifold of positive scalar curvature. Among the results we prove that if \(\Sigma_ 1\coprod...\coprod\Sigma_ n\subset S^ 3\) are mutually disjoint minimal surfaces embedded in a 3-sphere of positive scalar curvature, then there exists a decomposition \(S^ 3=S^ 3\#...\#S^ 3\) with each \(\Sigma_ j\) in a distinct summand. The technique used gives a new and more elementary proof of the fact that a compact minimal surface \(\Sigma\) embedded in a 3-sphere of positive scalar curvature must be topologically standard. We also present a series of examples of immersed hypersurfaces in the n-sphere \(S^ n\) which can never be minimal for any metric of positive scalar curvature on \(S^ n\).
scientific article; zbMATH DE number 3876146