具有有限总曲率子流形的L2调和p形式

L2-harmonic p-forms on submanifolds with finite total curvature

摘要: 设M是n+l维Sn+l球空间中具有法从平坦n维完备子流形，则Hp(L2(M))是M上L2调和p(2≤p≤n-2) 形式空间.首先证明了如果M的总曲率小于一个正常数，则Hp(L2(M))是平凡的;其次证明了如果M的总曲率有限，则Hp(L2(M))是有限维的.

Abstract: Let M be an n-dimensional complete submanifold with flat normal bundle in an (n+l)-dimensional sphere Sn+l. Let Hp(L2(M)) be the space of all L2-harmonic p-forms (2≤p≤n-2) on M. Firstly, we show that Hp(L2(M)) is trivial if the total curvature of M is less than a positive constant depending only on n. Secondly, we show that the dimension of Hp(L2(M)) is finite provided the total curvature of M is finite.