Curvature Functionals for Curves in the Equi-Affine Plane

TL;DR

This paper derives variational formulas for curvature functionals in the equi-affine plane and analyzes their critical points under specific constraints, advancing understanding of affine differential geometry.

Contribution

It provides the first detailed study of critical points of equi-affine curvature functionals under area constraints and full-affine arc-length.

Findings

Critical points characterized for equi-affine curvature integral under area constraint

Critical points identified for full-affine arc-length functional

General variational formula established for curvature functionals in the equi-affine plane

Abstract

After having given the general variational formula for the functionals indicated in the title, the critical points of the integral of the equi-affine curvature under area constraint and the critical points of the full-affine arc-length are studied in greater detail.