The unboundedness of certain minimal submanifolds of positively curved Riemannian spaces

✍ Scribed by Yoe Itokawa; Katsuhiro Shiohama

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 543 KB
• Volume: 11
• Category: Article
• ISSN: 0926-2245
• DOI: 10.1016/s0926-2245(99)00021-2

✦ Synopsis

Communicated by K. Fukaya

Received 17 March I998 fZhstmc,r: We prove that a minima1 immersion of a complete Riemannian manifold A4 into another complete noncompact Riemannian manifold N of positive curvature must have an unbounded image provided that M has scalar curvature bounded away from --oo. This extends the unboundedness theorems of Gromoll and Meyer for complete geodesics and of Galloway and Rodriguez for parabolic minima1 surfaces. Furthermore. we prove that in case M is of codimension I, only the Ricci curvature and not necessarily the full sectional curvature of the ambient space N need be positive in order for the same conclusion to hold.