等距嵌入下Gauss-Codazzi-Ricci系统的弱连续性
Weak Continuity of the Gauss-Codazzi-Ricci System for Isometric Embedding
摘要：
我们建立了Gauss-Coddazi-Ricci系统关于$p>2$的一致$L^p$ -有界解序列的等距嵌入的弱连续性，这意味着流形的等距嵌入的弱极限仍然是等距嵌入.更一般地，我们建立了Gauss-Codazzi-Ricci系统在微分几何中的补偿紧框架。也就是说，给定这个系统的任何序列的近似解是一致有界的$L^2$和有合理的界限的误差（误差被限制在一个紧凑的子集$H^{-1}_{\text{loc}}$），那么近似序列有一个弱收敛的子序列，其极限是一个解决方案的高斯Codazzi-里奇系统。此外，一个最小化的问题，提出了一个选择标准。对于这些，对黎曼曲率张量没有限制。
英文摘要：
We establish the weak continuity of the Gauss-Coddazi-Ricci system for isometric embedding with respect to the uniform $L^p$-bounded solution sequence for $p>2$, which implies that the weak limit of the isometric embeddings of the manifold is still an isometric embedding. More generally, we establish a compensated compactness framework for the Gauss-Codazzi-Ricci system in differential geometry. That is, given any sequence of approximate solutions to this system which is uniformly bounded in $L^2$ and has reasonable bounds on the errors made in the approximation (the errors are confined in a compact subset of $H^{-1}_{\text{loc}}$), then the approximating sequence has a weakly convergent subsequence whose limit is a solution of the Gauss-Codazzi-Ricci system. Furthermore, a minimizing problem is proposed as a selection criterion. For these, no restriction on the Riemann curvature tensor is made.
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