Hölder Parameterization of Continuous Quasi-Self-Contracted Curves in Complete Geodesic Spaces

In this paper, we introduce the notion of quasi-self-contracted curves (QSC curves for short) in metric spaces. It is a natural generalization of the notion of self-contracted curves, which was introduced by Daniilidis et al. (J Math Anal Appl 457(2):1333–1352, 2018,) to study gradient systems of quasi-convex functions. When the QSC constant c₀ equals 1, 1-QSC curves are exactly the self-contracted curves. It is well known (Daniilidis et al. in J Math Pures Appl 94(2):183–199, 2010, Lebedeva in Int Math Res Not 2021(11):8623–8656, 2020) that continuous self-contracted curves admit Lipschitz parameterization in many spaces. But continuous QSC curves do not in general, if the QSC constant c₀<1. We thus consider Hölder parameterization instead. We first show that any continuous QSC curve in any complete geodesic space X admits a Hölder parameterization if X supports a doubling measure. Then we investigate the case when c₀ is close to 1, and use a better estimate to show that the Hölder exponent also goes to 1 in a big class of metric spaces, i.e. complete CAT(0) spaces with some additional geometric properties.