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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.
Bulletin of the Brazilian Mathematical Society, New Series – Springer Journals
Published: Aug 12, 2017
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