Geodesic completeness for Sobolev metrics on the space of immersed plane curves (Q2879432)

Sobolev-type metrics on the space \(\text{Imm}(S^1, \mathbb{R}^2)\) of all smooth, immersed, closed, plane curves have interesting applications in computer vision, shape classification, and tracking. The applications are mainly related to the induced metrics on the \textsl{shape space}, i.e. the orbit space of \(\text{Imm}(S^1, \mathbb{R}^2)\) under the action of the reparametrization group. The idea of Sobolev-type metrics were introduced by \textit{P. W. Michor} and \textit{D. Mumford} [J. Eur. Math. Soc. (JEMS) 8, No. 1, 1--48 (2006; Zbl 1101.58005)].NEWLINENEWLINEIn the present paper it is proved that the geodesic equation for Sobolev-type metrics with constant coefficients of order 2 and higher is globally well-posed for smooth initial data as well as for initial data in certain Sobolov spaces. Consequently, the space \(\text{Imm}(S^1, \mathbb{R}^2)\) equipped with such a metric is geodesically complete.NEWLINENEWLINEThe details in the proof of this result are complicated and technical. However, all details are carefully explained and the paper is therefore very useful for someone who wants to enter into this promising area of research.