余维数为1的浸入到球面中的几何刚性估计
A geometric rigidity estimate for codimension-1 immersions into spheres
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中文总结 AI 辅助
本文针对球面中余维数一的浸入，建立了由拉伸加弯曲能量控制的定量刚性估计，无需先验基本形式界，并通过法向延伸与局部化拼接证明。
中文摘要 AI 辅助
我们建立了余维数为一的浸入到标准球面中的定量稳定性估计。设$(M,g)$是一个有向黎曼流形，并假设一个给定的形状算子由光滑等距浸入$\theta:(M,g)\to\mathbb S^{n+1}$实现。我们证明，在每个相对紧的强Lipschitz域上，且对每个$1<p<\infty$，任何Sobolev浸入$\phi$在模去一个环境旋转后都接近于$\theta$，其中浸入及其高斯映射之间的$W^{1,p}$距离由$\phi$的$L^p$拉伸加弯曲能量控制。不需要对$\phi$的基本形式进行先验界。证明过程通过沿法向测地线延伸浸入，并将问题归结为球面上的等维几何刚性估计。一个有限的局部化和拼接论证处理了参考浸入的法向延伸可能非单射的情况。
英文摘要
We establish a quantitative stability estimate for codimension-one immersions into a round sphere. Let $(M,g)$ be an oriented Riemannian manifold and suppose that a prescribed shape operator is realised by a smooth isometric immersion $θ:(M,g)\to\mathbb S^{n+1}$. We prove that, on every relatively compact strongly Lipschitz domain and for every $1<p<\infty$, any Sobolev immersion $ϕ$ is close to $θ$ modulo an ambient rotation, with the $W^{1,p}$-distances between both the immersions and their Gauss maps controlled by the $L^p$ stretching-plus-bending energies of $ϕ$. No {\it a priori} bounds on the fundamental forms of $ϕ$ are required. The proof proceeds by extending the immersions along normal geodesics and reducing the problem to an equidimensional geometric rigidity estimate on the sphere. A finite localisation and patching argument handles the possible non-injectivity of the normal extension of the reference immersion.
发表机构
- Mathematical Institute and Hertford College, University of Oxford（牛津大学数学研究所与赫特福德学院）
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