Curvature weighted metrics on shape space of hypersurfaces in $n$-space

Abstract: Let $M$ be a compact connected oriented $n-1$ dimensional manifold without boundary. In this work, shape space is the orbifold of unparametrized immersions from $M$ to $\mathbb R^n$. The results of \cite{Michor118}, where mean curvature weighted metrics were studied, suggest incorporating Gau{\ss} curvature weights in the definition of the metric. This leads us to study metrics on shape space that are induced by metrics on the space of immersions of the form $$ G_f(h,k) = \int_{M} \Phi . \bar g(h, k) \vol(f^*\bar{g}).$$ Here $f \in \Imm(M,\R^n)$ is an immersion of $M$ into $\R^n$ and $h,k\in C^\infty(M,\mathbb R^n)$ are tangent vectors at $f$. $\bar g$ is the standard metric on $\mathbb R^n$, $f^*\bar g$ is the induced metric on $M$, $\vol(f^*\bar g)$ is the induced volume density and $\Phi$ is a suitable smooth function depending on the mean curvature and Gau{\ss} curvature. For these metrics we compute the geodesic equations both on the space of immersions and on shape space and the conserved momenta arising from the obvious symmetries. Numerical experiments illustrate the behavior of these metrics.