Pure and Applied Mathematics Quarterly

Volume 9 (2013)

Number 2

Special Issue: In Honor of Dennis Sullivan, Part 2 of 2

Curvatures of Sobolev metrics on diffeomorphism groups

Pages: 291 – 332

DOI: http://dx.doi.org/10.4310/PAMQ.2013.v9.n2.a2

Authors

B. Khesin (Department of Mathematics, University of Toronto, Canada)

J. Lenells (Department of Mathematics, Baylor University, Waco, Texas, U.S.A.)

G. Misiolek (Institute for Advanced Study, Princeton, New Jersey, U.S.A.; Department of Mathematics, University of Notre Dame, Notre Dame, Indiana, U.S.A.)

S. C. Preston (Department of Mathematics, University of Colorado, Boulder, Co., U.S.A.)

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.

Keywords

Riemannian metrics, diffeomorphism groups, sectional curvature, stability, Euler-Arnold equations

2010 Mathematics Subject Classification

53C21, 58D05, 58D17

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