Abstract We present a system that builds 3D models of non-rigidly moving surfaces from scratch in real time using a single RGB-D stream. Our solution is based on the variational level set method, thus it copes with arbitrary geometry, including topological changes. It warps a given truncated signed distance field (TSDF) to a target TSDF via gradient flow. Unlike previous approaches that define the gradient using an L2 inner product, our method relies on gradient flow in Sobolev space. Its favourable regularity properties allow for a more straightforward energy formulation that is faster to compute and that achieves higher geometric detail, mitigating the over-smoothing effects introduced by other regularization schemes. In addition, the coarse-to-fine evolution behaviour of the flow is able to handle larger motions, making few frames sufficient for a high-fidelity reconstruction. Last but not least, our pipeline determines voxel correspondences between partial shapes by matching signatures in a low-dimensional embedding of their Laplacian eigenfunctions, and is thus able to reliably colour the output model. A variety of quantitative and qualitative evaluations demonstrate the advantages of our technique