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Riemannian geometries on spaces of plane curves. (English) Zbl 1101.58005
A compact simply connected region in the plane whose boundary is a simple closed curve is called a shape. In this paper, the authors consider the space of shapes viewed as the orbit space $$B_ e(S^ 1,\mathbb R^ 2) = \text{Emb}(S^ 1,\mathbb R^ 2)/\text{Diff}(S^ 1)$$ of embeddings or $$B_ i(S^ 1,\mathbb R^ 2) = \text{Imm}(S^ 1,\mathbb R^ 2)/\text{Diff}(S^ 1)$$ of immersions from $$S^ 1$$ to the plane modulo the group of diffeomorphisms of $$S^ 1$$, acting as reparametrizations. The goal of this paper is to find the simplest Riemannian metric on $$B_ e$$ or $$B_ i$$. For a constant $$A>0$$, the authors investigate the metric $$G^ A_ c(h,k)=\int_{S^1}(1+A\kappa_ c(\theta)^ 2)(h(\theta),k(\theta)| c'(\theta)| \,d\theta$$, where $$\kappa_ c$$ is the curvature of the curve $$c$$ and $$h$$, $$k$$ are normal vector fields to $$c$$. If $$A = 0$$, then the geodesic distance between any two curves is $$0$$. It is shown that the length function $$\ell : B_ e(S^ 1, \mathbb R^ 2)\to \mathbb R$$ satisfies $$\sqrt{\ell(C_ 1)}-\sqrt{\ell(C_ 2)}\leq \frac1{2\sqrt A}\text{dist}^{B_ e}_{G^ A}(C_ 1,C_ 2)$$. The authors find the geodesic equation for the metric $$G^ A$$ on $$\text{Emb}(S^ 1, \mathbb R^ 2)$$ and on $$B_ e(S^ 1, \mathbb R^ 2)$$. It is a non-linear partial differential equation of order $$4$$ with degenerate symbol. If $$A = 0$$, then the equation reduces to a non-linear second order hyperbolic PDE, which gives a well defined local geodesic spray. Also, the problem of existence and uniqueness of geodesics for $$A$$ is considered. For any $$A$$, the authors determine the sectional curvature on $$B_ e(S^ 1, \mathbb R^ 2)$$ and show that it is non-negative if $$A = 0$$ and negative otherwise. Finally, the authors solve the geodesic equation with simple endpoints numerically, and pose some open questions.

##### MSC:
 58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds 58D15 Manifolds of mappings 58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
##### Keywords:
Riemannian metrics
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##### References:
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