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| 摘要: |
| 本文给出了浸没在Hadamard流形N中的完备子流形M上的p-调和形式的一些消灭定理. 首先, 我们假设M满足加权庞加莱不等式且有平坦的法丛, 并进一步假设N具有纯曲率张量，对于 $2\leq l\leq n-2$，N的$(l,n-l)$-曲率不小于$-k\rho (0\leq k\leq\frac{4}{p^{2}})$, 如果总曲率很小，我们证明了p调和l-形式的一个消灭定理，它推广了Wang--Chao--Wu--Lv在[27]中的结果; 其次，假设N是一个具有截面曲率为$-k^{2}\leq K_{N}\leq 0$ 的Hadamard流形, 其中k为常数, 当总曲率足够小, 且拉普拉斯的第一特征值满足一定的下界时, 得到了p-调和1型的一个消灭定理, 推广Dung--Seo在[8]上的研究结果. |
| 关键词: p-调和形式，消灭定理，加权庞加莱不等式，Hadamard流形 |
| DOI： |
| 分类号:53C24, 53C21 |
| 基金项目: |
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| SOME VANISHING THEOREMS FOR p-HARMONIC FORMS ON SUBMANIFOLDS IN HADAMARD MANIFOLDS |
|
li nan
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| Abstract: |
| In this paper, we give some vanishing theorems for $p$-harmonic forms on a commplete submanifold $M$ immersed in Hadamard manifold $N$. Firstly, assume that $M$ satisfies the weighted Poincar\"e inequality and has flat normal bundle. And assume further that $N$ has pure curvature tensor and the $(l,n-l)$-curvature of $N$ is not less than $-k\rho (0\leq k\leq\frac{4}{p^{2}})$ for $2\leq l\leq n-2$.
If the total curvature is small enough, we prove a vanishing theorem for $p$-harmonic $l$-forms, which generalizes Wang--Chao--Wu--Lv"s results in [27]. Secondly, suppose that $N$ is a Hadamard manifold with sectional curvature $-k^{2}\leq K_{N}\leq 0$ for some constant $k$. If the total curvature is small enough and the first eigenvalue of Laplace satisfies a certain lower bound, we obtain a vanishing theorem for $p$-harmonic $1$-forms, which generalizes Dung--Seo"s results in [8]. |
| Key words: p-harmonic forms, vanishing theorems, weighted Poincaré inequality, Hadamard manifolds. |
