Hamiltonian Systems

The space of immersions of a closed manifold M into Euclidean space is among the most important infinite dimensional manifolds. 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”. They arise as natural extensions of right-invariant metrics on diffeomorphism groups, central to Arnold’s geometric formulation of hydrodynamical equations. Beyond their theoretical significance, they play a crucial role in mathematical shape analysis and geometric data science, where they enable meaningful and robust comparisons between shapes modeled as curves or surfaces.

In 2013, David Mumford conjectured that for orders larger than dim(M)/2 + 1, these geometries are complete. Note, that by the seminal work of Ebin and Marsden a similar statement is known to be true for diffeomorphism groups. In the context of immersions this conjecture was shown to be true in the case of immersed curves in the work of Bruveris, Michor and Mumford. 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.

Based on joint work with Cy Maor and Benedikt Wirth.

https://arxiv.org/abs/2512.01566