Sobolev metrics on shape space of surfaces

Martin Bauer; Philipp Harms; Peter W. Michor

doi:10.3934/jgm.2011.3.389

Article Contents

2011, Volume 3, Issue 4: 389-438. Doi: 10.3934/jgm.2011.3.389

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Sobolev metrics on shape space of surfaces

1.
Fakultät f¨ur Mathematik, Universität Wien, Nordbergstrasse 15, A-1090 Wien
2.
EdLabs, Harvard University, 44 Brattle Street, Cambridge, MA 02138

Received: September 2010

Revised: August 2011

Published: December 2011

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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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Mathematics Subject Classification: Primary: 58B20, 58D15, 58E12.

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