Sobolev Metrics on Shape Space of Surfaces in N-space

This paper extends parts of the results from [14] for plane curves to the case of surfaces in Rn. Let M be a compact connected oriented manifold of dimension less than n without boundary. Then shape space is either the manifold of submanifolds of Rn of type M , or the orbifold of immersions from M to Rn modulo the group of diffeomorphisms of M . We investigate the Sobolev Riemannian metrics on shape space: These are induced by metrics of the following form on the space of immersions: Gf (h, k) = ∫ M g(P h, k) vol(f∗ḡ) where g is the standard metric on Rn, f∗g is the induced metric on M , h, k ∈ C∞(M,Rn) are tangent vectors at f to the space of embeddings or immersions, and P f : C∞(M,Rn) → C∞(M,Rn) is a positive, selfadjoint, bijective scalar pseudo differetial operator of order 2p depending smoothly on f . We consider later specifically the operator P f = 1+A∆p, where ∆ is the Bochner-Laplacian on M induced by the metric f∗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 and give examples of numerical solutions.