Sobolev metrics on shape space of surfaces

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: $$ G^P_f(h,k) = \int_{M} \g(P^f h, k)\, \vol(f^*\g)$$ where $\g$ is some fixed metric on $N$, $f^*\g$ 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 $f^*\bar g$. 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.Comment: 52 pages, final version as it will appea

Fractional Sobolev metrics on spaces of immersions

“…The following corollary of our main result further generalizes this to fractional-order metrics on spaces of manifold-valued curves: The last special case to be discussed in this section is N = R n , which includes in particular the space of surfaces in R 3 . In the article [11] we proved a local wellposedness result for integer-order metrics. The proof given there had a gap, which has been corrected in the article [49].…”

“…(a) follows from the following formula for the first variation of the pull-back metric [11,Lemma 5.5]:…”

“…The treatment of general manifolds N requires a theory of Sobolev mappings between manifolds, which is developed in Section 2.2. Moreover, in the absence of global coordinate systems for these mapping spaces, we recast the geodesic equation using an auxiliary covariant derivative following [11]; see Lemma 2.6 and Theorem 4.3. • For M = N our result specializes to the diffeomorphism group Diff(M ),…”

“…Setting. We use the notation of [11] and write N for the natural numbers including zero. Smooth will mean C ∞ and real analytic C ω .…”

Exploring the Geometry of the Space of Shells

“…To avoid the resulting unphysical wrinkling effects, a supplementary (non physical) regularization was incorporated by Killian and co-workers. Instead one may use the regularizing effect of bending energy-and stay entirely in a physical simulation , who studied geodesic paths between surfaces parametrized over the unit sphere, using local changes of the area element as a Riemannian metric; Bauer et al [BHM11] investigated geodesic paths on the space of surfaces described by embeddings or immersions of a given manifold using as Riemannian metric a quadratic form corresponding to a higher order elliptic operator; finally Jin et al [JZLG09] studied Teichmüller space as a finite dimensional shape manifold, where shapes are classes of conformally equivalent surfaces. Our method falls into the category of shape space based approaches and we show rigorously that the energy Hessian of a simple class of stretching/bending models does indeed provide a proper Riemannian metric.…”