Sobolev Metrics on Shape Space of Hypersurfaces in N-space

This paper extends parts of the results from [10] for plane curves to the case of hypersurfaces in R. Let M be a compact connected oriented n − 1 dimensional manifold without boundary. Then shape space is either the manifold of submanifolds of R of type M , or the orbifold of immersions from M to R 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: GPf (h, k) = ∫ M ḡ(P h, k) vol(f∗ḡ) where ḡ is the standard metric on R, fḡ is the induced metric on M , h, k ∈ C(M,R) are tangent vectors at f to the space of embeddings or immersions, and where P f : C(M,R) → C(M,R) 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∆ where ∆ is the Bochner-Laplacian induced by the metric fḡ. 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.