Einstein four-manifolds of three-nonnegative curvature operator
Einstein four-manifolds of three-nonnegative curvature operator
复制标题
爱因斯坦三非负曲率算子的四流形
DOI：
10.1007/s00209-019-02296-8
复制
发表时间：
2019
影响因子：
0.8
通讯作者：
Wu Peng
中科院分区：
文献类型：
--
作者：
Wu Peng
本文证明了3-正曲率算子的爱因斯坦四流形是等距的，3-非负曲率算子的爱因斯坦四流形是等距的。我们还证明了具有非负截面曲率的爱因斯坦四流形的拉普拉斯算子的第一特征值在上有界。证明的基本思想是构造一个“积分次调和函数”，证明的主要成分是曲率分解（特别是Berger分解）、Weitzenböck公式和改进的加藤不等式。随着证明，我们还发现了使用Berger分解的Weitzenböck公式的替代证明，以及使用Derdziński的论证的改进Kato不等式的替代证明。
In this paper we prove that Einstein four-manifolds of 3-positive curvature operator are isometric toor, and Einstein four-manifolds of 3-nonnegative curvature operator are isometric to,, or, up to rescaling. We also prove that the first eigenvalue of the Laplace operator for Einstein four-manifolds withand nonnegative sectional curvature is bounded above by. The basic idea of the proofs is to construct an “integrated subharmonic function”, and the main ingredients of the proofs are curvature decompositions (in particular Berger decomposition), the Weitzenböck formula, and the refined Kato inequality. Along with the proofs, we also discover an alternative proof for the Weitzenböck formula using Berger decomposition, and an alternative proof for the refined Kato inequality using Derdziński’s argument.