We present a novel method to adaptively apply modifications to scene data stored in GPU memory. Such modifications may include interactive painting and sculpting operations in an authoring tool, or deformations resulting from collisions between scene objects detected by a physics engine. We only allocate GPU memory for the faces affected by these modifications to store fine-scale color or displacement values. This requires dynamic GPU memory management in order to assign and adaptively apply edits to individual faces at runtime. We present such a memory management technique based on a scan-operation that is efficiently parallelizable. Since our approach runs entirely on the GPU, we avoid costly CPU-GPU memory transfer and eliminate typical bandwidth limitations. This minimizes runtime overhead to under a millisecond and makes our method ideally suited to many real-time applications such as video games and interactive authoring tools. In addition, our algorithm significantly reduces storage requirements and allows for much higher-resolution content compared to traditional global texturing approaches. Our technique can be applied to various mesh representations, including Catmull-Clark subdivision surfaces, as well as standard triangle and quad meshes. In this paper, we demonstrate several scenarios for these mesh types where our algorithm enables adaptive mesh refinement, local surface deformations, and interactive on-mesh painting and sculpting.