Almost Local Metrics on Shape Space of Hypersurfaces in n-Space

This paper extends parts of the results from [17] for plane curves to the case of hypersurfaces in Rn. Let M be a compact connected oriented n − 1 dimensional manifold without boundary like S2 or the torus S1 × S1. 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 almost local Riemannian metrics on shape space: These are induced by metrics of the following form on the space of immersions: Gf (h, k) = ∫ M Φ(Vol(M),Tr(L))ḡ(h, k) vol(f∗ḡ) where ḡ is the standard metric on Rn, f∗ḡ is the induced metric on M , h, k ∈ C∞(M,Rn) are tangent vectors at f to the space of embeddings or immersions, where Φ : R2 → R>0 is a suitable smooth function, Vol(M) = ∫ M vol(f ∗ḡ) is the total hypersurface volume of f(M), and where the trace Tr(L) of the Weingarten mapping is the mean curvature. For these metrics we compute the geodesic equations both on the space of immersions and on shape space, the conserved momenta arising from the obvious symmetries, and the sectional curvature. For special choices of Φ we give complete formulas for the sectional curvature. Numerical experiments illustrate the behavior of these metrics.