Certain optimal correspondences between plane curves, I: Manifolds of shapes and bimorphisms

David Groisser
2009 Transactions of the American Mathematical Society  
In previous joint work, a theory introduced earlier by Tagare was developed for establishing certain kinds of correspondences, termed bimorphisms, between simple closed regular plane curves of differentiability class at least C 2 . A class of objective functionals was introduced on the space of bimorphisms between two fixed curves C 1 and C 2 , and it was proposed that one define a "best non-rigid match" between C 1 and C 2 by minimizing such a functional. In this paper we prove several
more » ... ove several theorems concerning the nature of the shape-space of plane curves and of spaces of bimorphisms as infinite-dimensional manifolds. In particular, for 2 ≤ j < ∞, the space of parametrized bimorphisms is a differentiable Banach manifold, but the space of unparametrized bimorphisms is not. Only for C ∞ curves is the space of bimorphisms an infinite-dimensional manifold, and then only a Fréchet manifold, not a Banach manifold. This paper lays the groundwork for a companion paper in which we use the Nash Inverse Function Theorem and our results on C ∞ curves and bimorphisms to show that if Γ is strongly convex, if C 1 and C 2 are C ∞ curves whose shapes are not too dissimilar (C j -close for a certain finite j) and if neither curve is a perfect circle, then the minimum of a regularized objective functional exists and is locally unique.
doi:10.1090/s0002-9947-08-04496-6 fatcat:xxoftsxrpjav3noehxxfhtwnty