$L^2$-harmonic $p$-forms on submanifolds with finite total curvature

Abstract

Let H p (L 2 (M )) be the space of all L 2 -harmonic p-forms (2 ≤ p ≤ n − 2) on complete submanifolds M with flat normal bundle in spheres. In this paper, we first show that H p (L 2 (M )) is trivial if the total curvature of M is less than a positive constant depending only on n. Second, we show that the dimension of H p (L 2 (M )) is finite if the total curvature of M is finite. The vanishing theorem is a generalized version of Gan-Zhu-Fang theorem and the finiteness theorem is an extension of Zhu-Fang theorem.