关于浸入开曲线的度量完备化
On the Metric Completion of Immersed Open Curves
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中文总结 AI 辅助
本文刻画了实值浸入开曲线空间在Sobolev型黎曼度量下的度量完备化，证明额外极限点与闭区间的$H^n$-微分同胚群同胚，否定了既有猜想并提出更精细猜想。
中文摘要 AI 辅助
近年来，带有Sobolev型重参数化不变黎曼度量的浸入曲线空间之完备性性质已被广泛研究。对于闭曲线空间，已知足够高阶的常系数度量是完备的，但对于开曲线则并非如此。本文中，我们刻画了实值浸入开曲线空间关于这些度量的度量完备化。我们证明了完备化中额外极限点的集合与闭区间的$H^n$-微分同胚群同胚。这一结果否定了(Bauer等, Ann. Sc. Norm. Super. Pisa Cl. Sci. (2023))中提出的关于完备化结构的问题，并针对多维环境空间的情形提出了一个更精细的猜想。
英文摘要
The completeness properties of spaces of immersed curves endowed with Sobolev-type reparametrization-invariant Riemannian metrics have been studied extensively in recent years. Whereas constant-coefficient metrics of large enough order on the space of closed curves are known to be complete, this is not the case for open curves. In this work, we characterize the metric completion of the space of real-valued immersed open curves with respect to these metrics. We show that the set of additional limit points in the completion is homeomorphic to $H^n$-diffeomorphisms of a closed interval. This result answers a question of the structure of the completion, posed in (Bauer et al., Ann. Sc. Norm. Super. Pisa Cl. Sci. (2023)), in the negative, and suggests a more refined conjecture on the structure in the case of multi-dimensional ambient space.
发表机构
- University of Toronto（多伦多大学）
- Einstein Institute of Mathematics, Hebrew University of Jerusalem（希伯来大学耶路撒冷分校爱因斯坦数学研究所）
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