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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)
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[2] Cervera, V., Mascar√≥, F., Michor, P. W.: The action of the diffeomorphism group on the space of immersions. Diff. Geom. Appl. 1 , 391-401 (1991) · Zbl 0783.58012 · doi:10.1016/0926-2245(91)90015-2
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