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Riemannian geometries on spaces of plane curves

Peter W. Michor, David Mumford (2006)

Journal of the European Mathematical Society

We study some Riemannian metrics on the space of smooth regular curves in the plane, viewed as the orbit space of maps from $S^{1}$ to the plane modulo the group of diffeomorphisms of $S^{1}$ , acting as reparametrizations. In particular we investigate the metric, for a constant $A > 0$ , $G_{c}^{A} (h, k) : = \int_{S^{1}} (1 + A κ_{c} {(θ)}^{2}) 〈 h (θ), k (θ) 〉 | c^{'} (θ) | d θ$ where $κ_{c}$ is the curvature of the curve $c$ and $h$ , $k$ are normal vector fields to $c$ . The term $A κ^{2}$ is a sort of geometric Tikhonov regularization because, for $A = 0$ , the geodesic distance between any two distinct curves is 0, while for $A > 0$ the...

Displaying 41 – 55 of 55

Riemannian geometries on spaces of plane curves

Self-intersection Points in classical area-minimizing surfaces.

Stratification of minimal surfaces, mean curvature flows, and harmonic maps.

Surfaces minimales et la construction de Calabi-Penrose

Surfaces of minimal area enclosing a given body in $ℝ^{3}$

The Dirichlet problem for harmonic maps from the disk into the euclidean n-sphere

The equivariant Dehn's lemma and loop theorem.

The existence of gaps in minimal foliations.

The second variation of area of minimal surfaces in four-manifolds.

The uniqueness for minimal surfaces in S3.

Über die endliche Lösbarkeit des Plateau-Problems in Riemannschen Mannigfaltigkeiten.

Über die Stetigkeit teilweise freier Minimalflächen.

Un problème spectral inverse sur les surfaces

Unstable Solutions of the Euler Equations of Certain Variational Problems.

Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms.

Currently displaying 41 – 55 of 55