Sobolev metrics on shape space of surfaces
Open Access
- 1 January 2011
- journal article
- Published by American Institute of Mathematical Sciences (AIMS) in Journal of Geometric Mechanics
- Vol. 3 (4), 389-438
- https://doi.org/10.3934/jgm.2011.3.389
Abstract
Let $M$ and $N$ be connected manifolds without boundary with $\dim(M) < \dim(N)$, and let $M$ compact. Then shape space in this work is either the manifold of submanifolds of $N$ that are diffeomorphic to $M$, or the orbifold of unparametrized immersions of $M$ in $N$. We investigate the Sobolev Riemannian metrics on shape space: These are induced by metrics of the following form on the space of immersions: $$ G^P_f(h,k) = \int_{M} \overline{g}( P^fh, k) vol (f^*\overline{g})$$ where $\overline{g}$ is some fixed metric on $N$, $f^*\overline{g}$ is the induced metric on $M$, $h,k \in \Gamma(f^*TN)$ are tangent vectors at $f$ to the space of embeddings or immersions, and $P^f$ is a positive, selfadjoint, bijective scalar pseudo differential operator of order $2p$ depending smoothly on $f$. We consider later specifically the operator $P^f=1 + A\Delta^p$, where $\Delta$ is the Bochner-Laplacian on $M$ induced by the metric $f^*\overline{g}$. For these metrics we compute the geodesic equations both on the space of immersions and on shape space, and also the conserved momenta arising from the obvious symmetries. We also show that the geodesic equation is well-posed on spaces of immersions, and also on diffeomorphism groups. We give examples of numerical solutions.
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This publication has 13 references indexed in Scilit:
- Properties of Sobolev-type metrics in the space of curvesInterfaces and Free Boundaries, Mathematical Analysis, Computation and Applications, 2008
- A metric on shape space with explicit geodesicsRendiconti Lincei, Matematica e Applicazioni, 2008
- An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approachApplied and Computational Harmonic Analysis, 2007
- Riemannian geometries on spaces of plane curvesJournal of the European Mathematical Society, 2006
- Geodesic flow on the diffeomorphism group of the circleCommentarii Mathematici Helvetici, 2003
- Computable Elastic Distances Between ShapesSIAM Journal on Applied Mathematics, 1998
- The Module Structure Theorem for Sobolev Spaces on Open ManifoldsMathematische Nachrichten, 1998
- The action of the diffeomorphism group on the space of immersionsDifferential Geometry and its Applications, 1991
- Pseudodifferential Operators in ℝnPublished by Springer Science and Business Media LLC ,1987
- Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaitsAnnales de l'institut Fourier, 1966