Chen Gui-Qiang G, Li Siran
Mathematical Institute, University of Oxford, Oxford, OX2 6GG UK.
J Geom Anal. 2018;28(3):1957-2007. doi: 10.1007/s12220-017-9893-1. Epub 2017 Aug 18.
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.
我们关注黎曼流形上高斯 - 科达齐 - 里奇（GCR）方程的整体弱刚性以及黎曼流形到欧几里得空间的相应等距浸入。我们开发了一种统一的内在方法来建立GCR方程和黎曼流形等距浸入的整体弱刚性，该方法独立于局部坐标，并对先前的局部结果和论证提供了进一步的见解。临界情况也已得到分析。为此，我们首先在黎曼流形上以内在方式重新表述具有散度 - 旋度结构的GCR方程，并开发一个全局的、内在版本的散度 - 旋度引理和其他非线性技术来处理流形上的整体弱刚性。特别地，建立了一个关于巴拿赫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内在散度 - 旋度引理作为特殊情况。建立了具有较低正则性的黎曼流形上全局等距浸入、嘉当形式主义和GCR方程之间的等价性。在此过程中，我们还证明了一个关于具有较低正则性的黎曼流形的、与嘉当形式主义相关的新的弱刚性结果，并将具有相应不同度量的黎曼流形的弱刚性结果进行了推广。
