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

Abstract

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