Sobolev metrics on shape space, II: Weighted Sobolev metrics and almost local metrics

Martin Bauer; Philipp Harms; Peter W. Michor

doi:10.3934/jgm.2012.4.365

Article Contents

2012, Volume 4, Issue 4: 365-383. Doi: 10.3934/jgm.2012.4.365

This issue Previous Article Next Article Semi-simple generalized Nijenhuis operators

Sobolev metrics on shape space, II: Weighted Sobolev metrics and almost local metrics

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

Received: September 2011

Revised: May 2012

Published: December 2012

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Abstract

In continuation of [7] we discuss metrics of the form $$ G^P_f(h,k)=\int_M \sum_{i=0}^p\Phi_i\big(Vol(f)\big)\ \bar{g}\big((P_i)_fh,k\big) vol(f^*\bar{g}) $$ on the space of immersions $Imm(M,N)$ and on shape space $B_i(M,N)=Imm(M,N)/{Diff}(M)$. Here $(N,\bar{g})$ is a complete Riemannian manifold, $M$ is a compact manifold, $f:M\to N$ is an immersion, $h$ and $k$ are tangent vectors to $f$ in the space of immersions, $f^*\bar{g}$ is the induced Riemannian metric on $M$, $vol(f^*\bar{g})$ is the induced volume density on $M$, $Vol(f)=\int_M vol(f^*\bar{g})$, $\Phi_i$ are positive real-valued functions, and $(P_i)_f$ are operators like some power of the Laplacian $\Delta^{f^*\bar{g}}$. We derive the geodesic equations for these metrics and show that they are sometimes well-posed with the geodesic exponential mapping a local diffeomorphism. The new aspect here are the weights $\Phi_i(Vol(f))$ which we use to construct scale invariant metrics and order 0 metrics with positive geodesic distance. We treat several concrete special cases in detail.

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

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