论文标题
$ l^p $ - 连接的捆绑和等距浸入$ w^{2，p} $的平滑性 - 规律性
Smoothability of $L^p$-connections on bundles and isometric immersions with $W^{2,p}$-regularity
论文作者
论文摘要
我们关注两个相互关联的问题：连接1形式的平稳性，其规律性低的束相关性具有平滑曲率2形式的束，以及存在较低规律性的等距沉浸液的存在。我们首先证明，如果$ω$是$ l^p $ - 连接$ 1 $ - 在矢量捆绑包上，封闭的riemannian $ n $ -n $ -manifold $ \ mathcal $ \ Mathcal {m \ natcal {m} $，带有小$ l^p $ -norm（$ p> n $），然后curvature $ 2 $ -form $ -form -form $ -Form $ \ nath $ can $ $ l^p _ {\ rm loc} $ - 通过相同曲率的平滑连接（不一定是等价）的拓扑。我们的证明是根据S. Mardare在具有$ l^p $ - 第二个基本形式的表面基本理论的工作中进行的，本质上是基本的，仅使用Hodge分解和固定点定理。然后将此结果应用于较低规律性的Riemannian歧管的等距浸入研究。我们重新审查存在$ w^{2，p} $的证明 - 与任意$ n $ and $ n $ and $ n $ and $ k $的薄弱解决方案以及对Gauss-codazzi-ricci-ricci-ricci-rickizi-rications Ass Aim Age Ass Aim Againding A Aim Againding A的弱点，\ Mathbf {M}^n \ to \ Mathbf {r}^{r}^{n+k} $具有薄弱的解决方案。 $ \ Mathcal {m} $上的指标，该$ w^{2，p} $ - 但没有$ c^\ infty $ -ismetric inmotecrics浸入。
We are concerned with two interrelated problems: smoothability of connection 1-forms with low regularity on bundles with prescribed smooth curvature 2-forms, and existence of isometric immersions with low regularity. We first show that if $Ω$ is an $L^p$-connection $1$-form on a vector bundle over a closed Riemannian $n$-manifold $\mathcal{M}$ with small $L^p$-norm ($p>n$) and smooth curvature $2$-form $\mathscr{F}$, then $Ω$ can be approximated in the $L^p_{\rm loc}$-topology by smooth connections of the same curvature (not necessarily gauge equivalent). Our proof, adapted from S. Mardare's work on the fundamental theory of surfaces with $L^p$-second fundamental form, is elementary in nature and uses only Hodge decomposition and fixed point theorems. This result is then applied to the study of isometric immersions of Riemannian manifolds with low regularity. We revisit the proof for the existence of $W^{2,p}$-isometric immersion $\mathcal{M}^n \to \mathbf{R}^{n+k}$ with arbitrary $n$ and $k$ given weak solutions to the Gauss--Codazzi--Ricci equations, aiming at elucidating some global vs. local issues, and also we provide a characterisation for metrics on $\mathcal{M}$ that admit $W^{2,p}$- but no $C^\infty$-isometric immersions.
