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

TL;DR

This paper introduces curvature-weighted metrics on the shape space of hypersurfaces in -space, deriving geodesic equations, conserved quantities, and demonstrating their properties through numerical experiments.

Contribution

It proposes new metrics based on Gaussian and mean curvature weights, computes their geodesic equations, and explores their geometric and numerical properties.

Findings

Derived explicit geodesic equations for curvature-weighted metrics.

Identified conserved momenta from symmetries of the metrics.

Numerical experiments illustrate the behavior of these curvature-weighted shape metrics.

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.