Abstract
We investigate self-contracted curves, arising as (discrete or continuous-time) gradient curves of quasi-convex functions, and their rectifiability (finiteness of the lengths) in Euclidean spaces, Hadamard manifolds, and \({\mathrm {CAT}}(0)\)-spaces. In the Hadamard case, we give a quantitative refinement of the original proof of the rectifiability of bounded self-contracted curves (in general Riemannian manifolds) by Daniilidis et al. Our argument leads us to a generalization to \({\mathrm {CAT}}(0)\)-spaces satisfying several uniform estimates on their local structures. Upon these conditions, we show the rectifiability of bounded self-contracted curves in trees, books, and \({\mathrm {CAT}}(0)\)-simplicial complexes.
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Acknowledgements
I would like to thank Miklós Pálfia for stimulating discussions. The author was supported in part by JSPS Grant-in-Aid for Scientific Research (KAKENHI) 15K04844.
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Ohta, Si. Self-contracted Curves in \({\mathrm {CAT}}(0)\)-Spaces and Their Rectifiability. J Geom Anal 30, 936–967 (2020). https://doi.org/10.1007/s12220-018-00126-7
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DOI: https://doi.org/10.1007/s12220-018-00126-7