## Certain optimal correspondences between plane curves, II: Existence, local uniqueness, regularity, and other properties

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- by David Groisser PDF
- Trans. Amer. Math. Soc.
**361**(2009), 3001-3030 Request permission

## Abstract:

This paper is a companion to the author’s paper (this volume), in which several theorems were proven concerning the nature, as infinite-dimensional manifolds, of the shape-space of plane curves and of spaces of certain curve-correspondences called*bimorphisms*. In Tagare, O’Shea, Groisser, 2002, a class of objective functionals, depending on a choice of cost-function $\Gamma$, 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 use the Nash Inverse Function Theorem to show that for strongly convex functions $\Gamma$, if $C_1$ and $C_2$ are $C^\infty$ curves whose shapes are not too dissimilar (specifically, are $C^j$-close for a certain finite $j$), and neither is a perfect circle, then the minimum of a certain regularized objective functional exists and is locally unique. We also study certain properties of the Euler-Lagrange equation for the objective functional, and obtain regularity results for “exact matches” (bimorphisms for which the objective functional achieves its absolute minimum value of 0) that satisfy a genericity condition.

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## Additional Information

**David Groisser**- Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611-8105
- Email: groisser@math.ufl.edu
- Received by editor(s): April 5, 2004
- Received by editor(s) in revised form: February 11, 2007
- Published electronically: December 23, 2008
- © Copyright 2008 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**361**(2009), 3001-3030 - MSC (2000): Primary 53A04, 49K15
- DOI: https://doi.org/10.1090/S0002-9947-08-04497-8
- MathSciNet review: 2485415