Curvatures of Sobolev metrics on diffeomorphism groups

TL;DR

This paper explores the sectional curvature of diffeomorphism groups related to Sobolev metrics, linking curvature signs to stability of solutions of associated PDEs, and provides detailed calculations and general results.

Contribution

It surveys current knowledge and provides detailed curvature calculations for Sobolev metrics, showing that sectional curvature generally varies in sign, impacting stability analysis.

Findings

Sectional curvature varies in sign for most Sobolev metrics.

Negative curvature correlates with instability, positive with stability.

Provides detailed curvature calculations for continuum mechanics equations.

Abstract

Many conservative partial differential equations correspond to geodesic equations on groups of diffeomorphisms. Stability of their solutions can be studied by examining sectional curvature of these groups: negative curvature in all sections implies exponential growth of perturbations and hence suggests instability, while positive curvature suggests stability. In the first part of the paper we survey what we currently know about the curvature-stability relation in this context and provide detailed calculations for several equations of continuum mechanics associated to Sobolev $H^0$ and $H^1$ energies. In the second part we prove that in most cases (with some notable exceptions) the sectional curvature assumes both signs.