球空间中子流形上L^p调和1-形式的消灭定理

The Vanishing Theorems of L^p Harmonic 1-Forms on Submanifolds in Spheres

设Mm(m≥3)是m+n维球空间S^m+n中的m维完备定向非紧子流形,考虑子流形Mm上的Lp(p≥2)调和1-形式的存在性问题.记Φ为子流形Mm的无迹张量,则Mm的全曲率定义为||Φ||L^m(M)=(∫M|Φ|^mdM)^1/m,其中dM表示Mm的体积元.首先,在子流形Mm的全曲率有正上界的假设条件下,特别地,该正上界的取值仅依赖于子流形Mm的维数m和p,使用Bochner公式及球空间中子流形Ricci曲率的下界估计和Sobolev不等式,并利用截断函数法和Lp条件,得到了子流形Mm上不存在非平凡的Lp调和1-形式,即Lp调和1-形式的消灭定理.其次,考虑逐点条件,假设子流形Mm的无迹张量Φ的最大模函数有正上界,该正上界的取值仅依赖于m,使用同样的方法,证明了Mm上不存在非平凡的Lp调和1-形式.

Let Mm(m≥3) be a complete noncompact submanifold in sphere S m+n. Studying the vanishing theorems of Lp(p≥2) harmonic 1-forms on Mm. Let Φ denote the traceless second fundamental form of Mm, then the total curvature of Mm be defined by ||Φ||L^m(M)=(∫M|Φ|^mdM)^1/m denotes the volume element of Mm. First, assume that the total curvature of Mm is less than a constant which only depends on the dimension of Mm and p, it shows that there is no nontrivial Lp harmonic 1-forms on Mm byusing Bochner formula, the bottom estimate of Ricci curvature and Sobolev inequality of Mm in spheres, themethod of cut off function and the condition of Lp. Second, assume that the maximal norm of Φ arebounded from above by a constant only depends on m, it shows that there is no nontrivial Lp harmonic 1-forms on Mm by using the same method.