The space of immersions of a closed manifold M into Euclidean space is an infinite dimensional manifold. The most natural Riemannian metrics on this space are reparametrization-invariant Sobolev metrics; these form a hierarchy of metrics, based on their order—the number of derivatives they "see". In 2013, David Mumford conjectured that for orders larger than dim(M)/2 + 1, some of these metrics are complete (a similar statement is known to be true for diffeomorphism groups). So far, this was verified only for immersed curves. In this talk I will present the first construction of complete metrics on immersions of two-dimensional surfaces, discuss the context and techniques, as well as possible extensions to higher dimensions. No knowledge in infinite dimensional geometry will be assumed. Based on joint work with Martin Bauer and Benedikt Wirth.