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p-Harmonic l-forms on Complete Noncompact Submanifolds in Sphere with Flat Normal Bundle

p-Harmonic l-forms on Complete Noncompact Submanifolds in Sphere with Flat Normal Bundle In this paper, we investigate a complete noncompact submanifold $$M^m$$ M m in a sphere $$S^{m+t}$$ S m + t with flat normal bundle. We prove that the dimension of the space of $$L^p$$ L p p-harmonic l-forms (when $$m\ge 4$$ m ≥ 4 , $$2\le l\le m-2$$ 2 ≤ l ≤ m - 2 and when $$m=3$$ m = 3 , $$l=2$$ l = 2 ) on M is finite if the total curvature of M is finite and $$m\ge 3$$ m ≥ 3 . We also obtain that there are no nontrivial $$L^p$$ L p p-harmonic l-forms on M if the total curvature is bounded from above by a constant depending only on m, p, l. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Brazilian Mathematical Society, New Series Springer Journals

p-Harmonic l-forms on Complete Noncompact Submanifolds in Sphere with Flat Normal Bundle

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References (35)

Publisher
Springer Journals
Copyright
Copyright © 2017 by Sociedade Brasileira de Matemática
Subject
Mathematics; Mathematics, general; Theoretical, Mathematical and Computational Physics
ISSN
1678-7544
eISSN
1678-7714
DOI
10.1007/s00574-017-0051-y
Publisher site
See Article on Publisher Site

Abstract

In this paper, we investigate a complete noncompact submanifold $$M^m$$ M m in a sphere $$S^{m+t}$$ S m + t with flat normal bundle. We prove that the dimension of the space of $$L^p$$ L p p-harmonic l-forms (when $$m\ge 4$$ m ≥ 4 , $$2\le l\le m-2$$ 2 ≤ l ≤ m - 2 and when $$m=3$$ m = 3 , $$l=2$$ l = 2 ) on M is finite if the total curvature of M is finite and $$m\ge 3$$ m ≥ 3 . We also obtain that there are no nontrivial $$L^p$$ L p p-harmonic l-forms on M if the total curvature is bounded from above by a constant depending only on m, p, l.

Journal

Bulletin of the Brazilian Mathematical Society, New SeriesSpringer Journals

Published: Aug 12, 2017

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