Abstract
We prove that a complete non-compact submanifold in a complete manifold of partially non-negative sectional curvature has only one end if the Sobolev inequality holds on it and if its total curvature is not very big by showing a Liouville theorem for harmonic maps and by using a existence theorem of constant harmonic functions with finite energy. We also generalize a result by Cao–Shen–Zhu saying that a complete orientable stable minimal hypersurface in a Euclidean space has only one end to submanifolds in manifolds of partially non-negative sectional curvature. Some related results about the structure of the same kind of submanifolds are also obtained.
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Wang, Q. Complete submanifolds in manifolds of partially non-negative curvature. Ann Glob Anal Geom 37, 113–124 (2010). https://doi.org/10.1007/s10455-009-9176-6
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DOI: https://doi.org/10.1007/s10455-009-9176-6