Metric compatibility and determination in complete metric spaces

Abstract

The local slope operator was introduced in De Giorgi et al. (Atti Accad Naz Lincei Rend Cl Sci Fis Mat Nat 68:180–187, 1980) to study gradient flow dynamics in metric spaces. This tool has now become a cornerstone in metric evolution equations (see, e.g., Ambrosio et al. (Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics. Birkhauser, 2008). Very recently, in Daniilidis and Salas (Proc Am Math Soc 150:4325–4333, 2022), it was established that Lipschitz inf-compact functions are uniquely determined by their local slope and critical values. Compactness played a paramount role in this result, ensuring in particular the existence of critical points. We hereby emancipate from this restriction and establish a determination result for merely bounded from below functions, by adding an assumption controlling the asymptotic behavior. This assumption is trivially fulfilled if f is inf-compact. In addition, our result is not only valid for the (De Giorgi) local slope, but also for the main paradigms of average descent operators as well as for the global slope, case in which the asymptotic assumption becomes superfluous. Therefore, the present work extends simultaneously the metric determination results of Daniilidis and Salas (Proc Am Math Soc 150:4325–4333, 2022) and Thibault and Zagrodny (Commun Contemp Math 25(7):2250014, 2023).