Completeness of reparametrization-invariant Sobolev metrics on the space of surfaces

TL;DR

This paper extends the understanding of completeness properties of reparametrization-invariant Sobolev metrics from curves to surfaces, establishing conditions for metric and geodesic completeness and the existence of minimizing geodesics.

Contribution

It provides the first completeness results for immersed surfaces under Sobolev metrics, confirming a conjecture of Mumford and generalizing previous curve-based results.

Findings

Conditions for metric and geodesic completeness established

Existence of minimizing geodesics proven for certain Sobolev metrics

Validation of Mumford's conjecture on surface immersions

Abstract

We study reparametrization-invariant Sobolev-type Riemannian metrics on the space of immersed surfaces and establish conditions ensuring metric and geodesic completeness as well as the existence of minimizing geodesics. This provides the first extension of completeness results for immersed curves, originating from works of Bruveris, Michor, and Mumford, and validates an earlier conjecture of Mumford on completeness properties of general spaces of immersions in this important case. The result is obtained by recasting earlier approaches to completeness on manifolds of mappings as a general completeness criterion for infinite-dimensional Riemannian manifolds that are open subsets of a complete Riemannian manifold and by combining it with geometric estimates based on the Michael--Simon--Sobolev inequality to establish the completeness for specific Sobolev metrics on immersed surfaces.