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Some results on the occurrence of compact minimal submanifolds

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Abstract

Varying the situation considered in Myers theorem, we show, via standard index form techniques, that a complete Riemannian manifold which admits a compact minimal submanifold is necessarily compact, provided a suitable curvature object is positive on the average along the geodesies issuing orthogonally from the minimal submanifold. By slightly recasting this result, one establishes the nonexistence of compact minimal submanifolds (in particular, closed geodesies) in complete noncompact manifolds which obey an appropriate curvature condition. A generalization of a result of Tipler concerning the occurrence of zeros of solutions to the scalar Jacobi equation is also obtained.

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  1. Department of Mathematics, University of Miami, 33124, Coral Gables, FL, USA

    Gregory J. Galloway

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  1. Gregory J. Galloway
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Galloway, G.J. Some results on the occurrence of compact minimal submanifolds. Manuscripta Math 35, 209–219 (1981). https://doi.org/10.1007/BF01168457

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