The unboundedness of certain minimal submanifolds of positively curved Riemannian spaces
We prove that a minimal immersion of a complete Riemannian manifold \(M\) into another complete noncompact Riemannian manifold \(N\) of positive curvature must have an unbounded image provided that \(M\) has scalar curvature bounded away from \(- \infty\). This extends the unboundedness theorems of \textit{D. Gromoll} and \textit{W. Meyer} [Ann. Math., II. Ser. 90, 75-90 (1969; Zbl 0191.19904)] for complete geodesics and of \textit{G. J. Galloway} and \textit{L. Rodriguez} [Geom. Dedicata 39, 29-42 (1991; Zbl 0724.53037)] for parabolic minimal surfaces. Furthermore, we prove that in case \(M\) is of codimension 1, only the Ricci curvature and not necessarily the full sectional curvature of the ambient space \(N\) needs to be positive in order for the same conclusion to hold.
scientific article; zbMATH DE number 1367707
scientific article