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Global Weak Rigidity of the Gauss-Codazzi-Ricci Equations and Isometric Immersions of Riemannian Manifolds with Lower Regularity.
The Journal of Geometric Analysis ( IF 1.1 ) Pub Date : 2017-08-18 , DOI: 10.1007/s12220-017-9893-1
Gui-Qiang G Chen 1 , Siran Li 1
Affiliation  

We are concerned with the global weak rigidity of the Gauss–Codazzi–Ricci (GCR) equations on Riemannian manifolds and the corresponding isometric immersions of Riemannian manifolds into the Euclidean spaces. We develop a unified intrinsic approach to establish the global weak rigidity of both the GCR equations and isometric immersions of the Riemannian manifolds, independent of the local coordinates, and provide further insights of the previous local results and arguments. The critical case has also been analyzed. To achieve this, we first reformulate the GCR equations with div-curl structure intrinsically on Riemannian manifolds and develop a global, intrinsic version of the div-curl lemma and other nonlinear techniques to tackle the global weak rigidity on manifolds. In particular, a general functional-analytic compensated compactness theorem on Banach spaces has been established, which includes the intrinsic div-curl lemma on Riemannian manifolds as a special case. The equivalence of global isometric immersions, the Cartan formalism, and the GCR equations on the Riemannian manifolds with lower regularity is established. We also prove a new weak rigidity result along the way, pertaining to the Cartan formalism, for Riemannian manifolds with lower regularity, and extend the weak rigidity results for Riemannian manifolds with corresponding different metrics.

中文翻译:

Gauss-Codazzi-Ricci方程的整体弱刚度和具有较低规则性的黎曼流形的等距浸入。

我们关注在黎曼流形上的高斯-科达兹-里奇(GCR)方程的整体弱刚度以及黎曼流形到欧几里得空间中的相应等距浸入。我们开发了一种统一的内在方法,以建立GCR方程和黎曼流形的等距浸入的全局弱刚度,而与局部坐标无关,并提供对先前局部结果和参数的进一步见解。关键情况也已分析。为此,我们首先在黎曼流形上用div-curl结构重新构造GCR方程,并开发div-curl引理和其他非线性技术的全局固有版本,以解决流形上的全局弱刚度。特别是,建立了关于Banach空间的一般泛函分析补偿紧致性定理,其中包括黎曼流形上的固有div-curl引理。建立了具有较低规则性的黎曼流形上的整体等距沉浸,Cartan形式主义和GCR方程的等价关系。我们还证明了,对于具有较低规则性的黎曼流形,在此过程中也证明了与Cartan形式主义有关的新的弱刚度结果,并扩展了具有不同度量的黎曼流形的弱刚度结果。建立了具有较低规则性的黎曼流形上的GCR方程。我们还证明了,对于具有较低规则性的黎曼流形,在此过程中也证明了与Cartan形式主义有关的新的弱刚度结果,并扩展了具有不同度量的黎曼流形的弱刚度结果。建立了具有较低规则性的黎曼流形上的GCR方程。我们还证明了,对于具有较低规则性的黎曼流形,在此过程中也证明了与Cartan形式主义有关的新的弱刚度结果,并扩展了具有不同度量的黎曼流形的弱刚度结果。
更新日期:2017-08-18
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