Centro–affine curvature flows on centrally symmetric convex curves

Abstract:

We consider two types of $p$-centro-affine flows on smooth, centrally symmetric, closed convex planar curves: $p$-contracting and $p$-expanding. Here $p$ is an arbitrary real number greater than $1$. We show that, under any $p$-contracting flow, the evolving curves shrink to a point in finite time and the only homothetic solutions of the flow are ellipses centered at the origin. Furthermore, the normalized curves with enclosed area $\pi$ converge, in the Hausdorff metric, to the unit circle modulo $SL(2)$. As a $p$-expanding flow is, in a certain way, dual to a contracting one, we prove that, under any $p$-expanding flow, curves expand to infinity in finite time, while the only homothetic solutions of the flow are ellipses centered at the origin. If the curves are normalized to enclose constant area $\pi$, they display the same asymptotic behavior as the first type flow and converge, in the Hausdorff metric, and up to $SL(2)$ transformations, to the unit circle. At the end of the paper, we present a new proof of the $p$-affine isoperimetric inequality, $p\geq 1$, for smooth, centrally symmetric convex bodies in $\mathbb {R}^2$.