On Conformal Flexibility of Completeness and Minimizing Geodesics on Hilbert Manifolds

Abstract

This article is part of a broader programme investigating which features of finite-dimensional Riemannian geometry persist in infinite dimensions. The Hopf--Rinow theorem fails even for Hilbert manifolds: metric and geodesic completeness need not agree, and neither property guarantees a length-minimizing geodesic between prescribed endpoints. Despite this failure, our first main result shows that the conformal class of every smooth strong Riemannian metric on a smooth separable Hilbert manifold contains a smooth strong representative that is metrically and geodesically complete and such that every two points in the same connected component are joined by a length-minimizing geodesic. By contrast, our second main result establishes a local flexibility phenomenon for metric completeness that is inherently infinite-dimensional. Given any prescribed Hilbert-norm ball, a metrically complete strong metric admits a conformal deformation which is equal to one outside that ball, preserves geodesic completeness, and destroys metric completeness.