Mobile Augmented Reality (MAR) apps may cause short battery life due to high-quality virtual objects rendered in the augmented environment. State-of-the-art solutions propose to balance energy consumption and user-experience using a static set of decimated object versions within the app. However, they do not consider that each object has unique characteristics, which highly influence how the user-perceived quality changes according to user-object distance and triangle count. As a result, they may lead to limited energy savings, a high storage overhead, and a high burden on the MAR app developer. In this paper, we propose eAR, an edge-assisted autonomous and energy-efficient framework for MAR apps designed to solve the limitations of state-of-the-art solutions. eAR features an offline software running on an edge server that leverages Image Quality Assessment (IQA) to model user-perceived quality for each virtual object in terms of triangle count and user-object distance. In addition, eAR features a runtime lightweight optimization algorithm that dynamically decides the most energy-efficient virtual object triangle count to request from the edge server based on (i) the per-object models of user-perceived quality, (ii) energy consumption models for mobile GPU and network interface, and (iii) a user path prediction system that estimates near-future user-object distances. eAR is completely autonomous and can be easily integrated into most MAR apps as an open-source library. Our results show that eAR can help reduce energy consumption by up to 16.5% while reducing storage overhead by almost 60% compared to existing schemes, with minimal MAR app developer effort and minimal impact on user-perceived quality.