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Jamming is a fundamental transition that governs the mechanical behavior of particulate media, including sand, foam and dense suspensions but also biological tissues: Upon compression, particulate media can change from freely flowing to a disordered solid. Jamming has previously been conceived as a bulk phenomenon involving particle motions in fixed geometries. In a diverse class of soft materials, however, solidification can take place in a deformable geometry, such as on the surface of a fluid droplet or in the formation of a bijel. In these systems the nature and dynamics of jamming remains unknown. Here we propose and study a scenario we call metric jamming that aims to capture the complex interactions between shape and particles. Unlike classical jamming processes that exhibit discrete mechanical transitions, surprisingly we find that metric jammed states possess mechanical properties that are continuously tunable between those of classical jammed and conventional elastic media. New types of vibrational mode that couple particulate and surface degrees of freedom are also observed. Our work lays the groundwork for a unified understanding of jamming in deformable geometries, to exploit jamming for the control and stabilization of shape in self-assembly processes, and provides new tools to interrogate solidification processes in deformable media more generally.