论文标题
等距扩展的刚度和灵活性
Rigidity and Flexibility of Isometric Extensions
论文作者
论文摘要
在本文中，我们考虑了$ c^{1，θ} $等距扩展的刚度和灵活性，我们表明hölder指数$θ_0= \ frac12 $在下面的意义上至关重要：如果$ u \ in c^{1，θ} $是平稳的等值线的$ subman $ subman $ subman $ subman $ submanifting n sugimmanifting ossman in \ frac12 $，然后切向连接沿$σ$的Levi-Civita连接一致。另一方面，对于任何$θ<\ frac12 $，我们都可以通过违反此类属性的凸集成来构建$ c^{1，θ} $等距扩展。作为副产品，我们将$ c^{1，θ} $等速线嵌入，$θ<\ frac12 $的存在定理，$ c^1 $ c^1 $量级和敏感量的$θ<\ frac12 $。
In this paper we consider the rigidity and flexibility of $C^{1, θ}$ isometric extensions and we show that the Hölder exponent $θ_0=\frac12$ is critical in the following sense: if $u\in C^{1,θ}$ is an isometric extension of a smooth isometric embedding of a codimension one submanifold $Σ$ and $θ> \frac12$, then the tangential connection agrees with the Levi-Civita connection along $Σ$. On the other hand, for any $θ<\frac12$ we can construct $C^{1,θ}$ isometric extensions via convex integration which violate such property. As a byproduct we get moreover an existence theorem for $C^{1, θ}$ isometric embeddings, $θ<\frac12$, of compact Riemannian manifolds with $C^1$ metrics and sharper amount of codimension.
